Rutger's Formula

Rutger's formula relates the discontinuity of the specific heat of a material at the superconducting-critical temperature $T_{C}$ with its superconducting-critical field $H_{C}$. It is given by

$c_{s}-c_{n}=T\mu _{0}\left[{\frac {\mathrm {d} H_{C}(T)}{\mathrm {d} T}}\right]^{2}{\Big |}_{T=T_{C}}$

where $T$ is the temperature, $c_{s}$ is the specific heat of the material in the superconducting state (right before phase transition), $c_{n}$ is the specific heat of the material in the normal state (right before phase transition), and $\mu _{0}$ is the Vacuum permeability.

The equation can be obtained as follows: from the Meissner effect, we know that $B=0$ for $H<H_{C}$, so that the free energy density in a superconductor is

$F_{s}(H)=F_{s}(0)+\mu _{0}\int _{0}^{H}H\,\mathrm {d} H+{\frac {1}{2}}\mu _{0}H^{2}$

on the other hand, normal states are weakly magnetic or not magnetic at all, so that $F_{n}(H)=F_{n}(0)$. Furthermore, the free energy is a continuous function, so $F_{n}(H_{C})=F_{s}(H_{C})$, and so

$\Delta F=F_{s}(H_{C})-F_{s}(0)=F_{n}-F_{s}={\frac {1}{2}}\mu _{0}H_{c}^{2}$

Multiplying by the sample's volume $V$ and differentiating twice with respect to T, we get

$\Delta C=TV\mu _{0}\left\{\left[{\frac {\mathrm {d} H_{C}(T)}{\mathrm {d} T}}\right]^{2}+H_{c}{\frac {\mathrm {d} ^{2}H_{C}(T)}{\mathrm {d} T^{2}}}\right\}$

or

$\Delta c=c_{s}-c_{n}=T\mu _{0}\left\{\left[{\frac {\mathrm {d} H_{C}(T)}{\mathrm {d} T}}\right]^{2}+H_{c}{\frac {\mathrm {d} ^{2}H_{C}(T)}{\mathrm {d} T^{2}}}\right\}$

setting $T=T_{C}$, at which point $H_{C}=0$, we obtain

$c_{s}-c_{n}=T\mu _{0}\left[{\frac {\mathrm {d} H_{C}(T)}{\mathrm {d} T}}\right]^{2}{\Big |}_{T=T_{C}}$

which is Reutger's formula.

References

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