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We report on specific heat ($C_p$), transport, Hall probe and penetrationdepth measurements performed on Fe(Se$_{0.5}$Te$_{0.5}$) single crystals ($T_c\sim 14$ K). The thermodynamic upper critical field $H_{c2}$ lines has beendeduced from $C_p$ measurements up to 28 T for both $H\|c$ and $H\|ab$, andcompared to the lines deduced from transport measurements (up to 55 T in pulsedmagnetic fields). We show that this {\it thermodynamic} $H_{c2}$ line presentsa very strong downward curvature for $T \rightarrow T_c$ which is not visiblein transport measurements. This temperature dependence associated to an upwardcurvature of the field dependence of the Sommerfeld coefficient confirm that$H_{c2}$ is limited by paramagnetic effects. Surprisingly this paramagneticlimit is visible here up to $T/T_c \sim 0.99$ (for $H\|ab$) which is theconsequence of a very small value of the coherence length $\xi_c(0) \sim 4 \AA$(and $\xi_{ab}(0) \sim 15 \AA$), confirming the strong renormalisation of theeffective mass (as compared to DMFT calculations) previously observed in ARPESmeasurements [Phys. Rev. Lett. 104, 097002 (2010)]. $H_{c1}$ measurements leadto $\lambda_{ab}(0) = 430 \pm 50$ nm and $\lambda_c(0) = 1600 \pm 200$ nm andthe corresponding anisotropy is approximatively temperature independent ($\sim4$), being close to the anisotropy of $H_{c2}$ for $T\rightarrow T_c$. Thetemperature dependence of both $\lambda$ ($\propto T^2$) and the electroniccontribution to the specific heat confirm the non conventional couplingmechanism in this system.

Thermodynamic phase diagram of Fe(Se$_{0.5}$Te$_{0.5}$) single crystals up to 28 Tesla