= Oscillatory thermal-change amplitude ratio
{title2=$\mathcal R=6|\alpha|(1-X/\sinh X)/X^2$}
For the sinusoidal density-of-states model, divide the oscillatory <amplitude> of $\Phi(T,\mu)-\Phi(0,\mu)$ by its smooth magnitude $g_0\pi^2(k_BT)^2/6$. This gives the title ratio. Its limit as $T\to0$ is $|\alpha|$, although both changes vanish at exactly zero <temperature>. The expansion is $|\alpha|[1-7X^2/60+O(X^4)]$, and its large-$X$ envelope is $6|\alpha|/X^2$. A fixed <chemical potential> adds the actual <cosine> <phase>, which can make the oscillatory contribution zero even when the envelope is nonzero. Exponential suppression of the full oscillatory potential therefore must not be confused with this algebraic thermal-change ratio.
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