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Counting function for a sphere of anisotropic quartz.

Niels Søndergaard1, Thomas Guhr, Mark Oxborrow

  • 1Matematisk Fysik, LTH, Lunds Universitet, Sweden. Niels.Sondergaard@matfys.lth.se

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 5, 2004
PubMed
Summary

We numerically evaluated the Weyl term for quartz spheres, accounting for anisotropy. Our findings, derived using Radon transforms, align with experimental resonance data.

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Area of Science:

  • Solid State Physics
  • Materials Science
  • Mathematical Physics

Background:

  • Counting functions are essential for understanding quantum systems.
  • Previous studies often simplified material properties, neglecting anisotropy.
  • Monocrystalline quartz exhibits significant anisotropic behavior.

Purpose of the Study:

  • To calculate the leading Weyl term of the counting function for a monocrystalline quartz sphere.
  • To incorporate the anisotropic properties of quartz into the calculation.
  • To compare numerical results with experimental data.

Main Methods:

  • Numerical evaluation of the counting function.
  • Utilizing the Radon transform representation of the Green's function.
  • Accounting for the anisotropy of monocrystalline quartz.

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Main Results:

  • The anisotropy of quartz prevents a simple analytical form for the counting function.
  • A numerical evaluation was performed, yielding specific Weyl term values.
  • Results show good agreement with a unique, large experimental dataset of resonances.

Conclusions:

  • Anisotropic effects are critical for accurate counting function calculations in materials like quartz.
  • Numerical methods, particularly those employing Radon transforms, are effective for complex systems.
  • The study validates theoretical calculations against experimental measurements.