A Practical Approach to Specific Heat-Phonon Spectrum Inversion

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Thomas C. Butler and Dr. William E. Evenson, Physics and Astronomy

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.

Our approach relied heavily on the theoretical work of X.X. Dai who through explicit construction of the general solution provided necessary and sufficient conditions for the existence and uniqueness of the phonon spectrum from corresponding specific heat data.2 He also provided hints as to the direction one could take to regularize the solution. We were able to rework the theory Dai developed using simpler analysis and direct integration and also were able to demonstrate more clearly the instability of specific heat-phonon spectrum inversion (SPI). These results were surprises and are of secondary importance as they are simply accessible derivations of known results. However, they will make the work more accessible to future student researchers. This theoretical work ensured that a unique result can be obtained from data that has been well fitted to a smooth analytical expression.

With the theoretical groundwork laid, an expansion of the phonon spectrum in terms of odd Hermite functions was inserted in the specific heat equation. The expansion was over the unknown phonon spectrum, so the problem became to obtain the first several expansion coefficients for the phonon spectrum expansion. This expansion led to a set of fitting functions for specific heat that had the desired expansion coefficients for fitting parameters. Not surprisingly the fit was ill-conditioned, and we had to use the well known method of Singular Value Decomposition to stabilize calculations with the ill-conditioned matrices that resulted from the data. Scaled methods using dimensionless versions of the theory and fitted data were also essential to successful inversion.

We tested our method on several theoretical models and on YBCO. A striking example is an inversion of a calculated specific heat from a linear combination of two Debye spectra (Fig. 1). Most significant is the inversion of YBCO from real data. This was compared to the experimental phonon spectrum with remarkably good agreement for a severely ill posed problem, and serves as proof that the method can be useful for obtaining phonon spectra from real data with a useful level of accuracy (Fig 2.). This opens many new avenues of research into materials properties, as diverse phenomena are connected to the phonon spectrum, such as superconductivity, thermal expansion and many other interesting effects.

I had a wonderful experience researching this problem. The work is also are making an impact. At the regional American Physical Society meeting in October, I was awarded a best student presentation award. More importantly, I have received requests for an article from scientists at the conference. My mentor and I are in the final drafts of a paper to be submitted to Physical Review E, and are actively pursuing the application of this method to negative thermal expansion materials. This project introduced me to the excitement of scientific research and gave me confidence that my intended career in research science is well within my abilities. Figure 1: Blue dashes are theoretical spectrum for a linear combination of Debye model specific heats. The red line is calculated by inverting the simulated specific heat data. Figure 2: Blue dots are an experimental phonon spectrum for YBCO, red line is the phonon spectrum from inverting specific heat data.

References

  • H. Grayson-Smith, J. P. Stanley, Note on the Derivation of the Frequency Spectrum of a Crystal from Specific Heat Measurements. 236 (1949).
  • X. X. Dai, X. Xu, J. Dai, On a specific heat-phonon spectrum inversion problem. Exact solution, unique existence theorem and Riemann hypothesis. Phys Lett. A 147, 445 (1990).