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

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 $T>\mathrm{}{T}_a$, where $T_a=\frac{\hslash {\omega }_L}{2\pi k}$ and ${\omega }_L$ is the maximum normal mode frequency, and because of slow convergence, it is useless from a numerical point of view for $T<\mathrm{}\frac{4T_a}{3}$. The range of convergence of the expansion can be extended to absolute zero and its computational usefulness down to $T\approx \frac{2T_a}{3}$ by means of an Euler transformation, which effectively converts it into an expansion in $\frac{1}{(T^2+{T_b}^2)}$ with $T_b\approx T_a$. 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.