Thermal damping of a sinusoidal density of states ($R_T=X/\sinh X,X=2{\pi }^2{k}_{B}T/\Delta$)

A density of states $g(ϵ)=g_0[1+\alpha \cos (2\pi ϵ/\Delta )]$ has zero-temperature oscillatory grand potential $g_0\alpha {\Delta }^2\cos (2\pi \mu /\Delta )/(4{\pi }^2)$. Its Fermi-window convolution multiplies that oscillation by $R_T$ when the lower band edge is far below the chemical potential. The Fourier transform of the even window removes its sine component and gives the title factor. At high $X$, $R_T\sim 2X{e}^{-X}$; at small $X$, $R_T=1-X^2/6+O(X^4)$. The corresponding thermal change relative to zero temperature has amplitude proportional to $1-R_T$, rather than $R_T$.

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.