= Lattice qualification of an electron-only heat-capacity ratio
An electron-only comparison omits the lattice contribution to a metal's measured <heat capacity>. In a simple low-<temperature> normal metal, the same sample normalization gives
$$
C_{\rm metal}\simeq\gamma T+\beta T^3,
$$
where the linear term is <electronic heat capacity> and the cubic term is the leading <Debye model> lattice contribution. Both coefficients must use compatible units and amounts of material.
If a reference gas has <heat capacity> $C_{\rm gas}$, its ratio to the total metal value is
$$
\frac{C_{\rm gas}}{C_{\rm metal}}=\frac{C_{\rm gas}}{\gamma T}\frac1{1+\beta T^2/\gamma}.
$$
The electron-only ratio is therefore reduced when lattice <phonons> contribute appreciably. The crossover estimate is $T=\sqrt{\gamma/\beta}$, where the two terms are equal. <Superconductivity>, magnetic excitations or other low-energy structure can add further terms. A ratio that is valid for electronic degrees of freedom alone must consequently be distinguished from a ratio of whole-sample calorimetric measurements.
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