Extension of the Range of Validity of Thirring's Expansion for the Specific Heat of Crystals

Abstract

In 1913, Thirring obtained an expansion for the vibrational contribution to the specific heat of a crystalline solid in powers of $\frac{1}{{T}^{2}}$. The coefficients of this series are proportional to successive moments of the frequency spectrum. In its original form, Thirring's expansion converges only for $Tg{T}_{a}$, where ${T}_{a}=\frac{\ensuremath{\hbar}{\ensuremath{\omega}}_{L}}{2\ensuremath{\pi}k}$ and ${\ensuremath{\omega}}_{L}$ is the maximum normal mode frequency, and because of slow convergence, it is useless from a numerical point of view for $Tl\frac{4{T}_{a}}{3}$. The range of convergence of the expansion can be extended to absolute zero and its computational usefulness down to $T\ensuremath{\approx}\frac{2{T}_{a}}{3}$ by means of an Euler transformation, which effectively converts it into an expansion in $\frac{1}{({T}^{2}+{{T}_{b}}^{2})}$ with ${T}_{b}\ensuremath{\approx}{T}_{a}$. The improvement in convergence is so efficient that, usually, only the first 6 or 7 even moments are required to obtain four-figure accuracy at $T={T}_{a}$. Alternatively, nonlinear transformations can be applied if the specific heat is to be calculated for a few values of temperature only. Some examples of the use of these methods are presented. Conversely, Euler's transformation provides a means for a more detailed description of the frequency distribution from specific heat measurements.