 Oscillatory thermal-change amplitude ratio

For the sinusoidal density-of-states model, divide the oscillatory amplitude of $\Phi (T,\mu )-\Phi (0,\mu )$ by its smooth magnitude $g_0{\pi }^2(k_{B}T{)}^2/6$. This gives the title ratio. Its limit as $T\to 0$ 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.