4 while some deviations between the calculated and observed specific heat can be seen in Fig. 2. Some such deviations are expected because feedback effects of superconductivity on the bosonic spectral function, which lead to the formation of a temperature-dependent “resonant mode” in the spin fluctua- tion spectrum below T$_{c}$ [27], have not been considered in the calculations. In summary, the specific heat of hole-doped Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ was measured on crystals of supe- rior quality and phase purity. Our α -model fit to the electronic part of the specific heat shows that there are two groups of superconducting gaps with magnitudes of 11 and 3.5 meV. The magnitude and shape of the superconducting jump anomaly indicate strong coupling with intermediate bosons, in good agreement with an Eliashberg analysis of a spin-fluctuation model. We thank L. Boeri and A. N. Yaresko for helpful discussion and G. Siegle for experimental assistance. We acknowledge support by the Deutsche Forschungsgemeinschaft (DFG) via grant BO 3537/1-1 in SPP 1458. FIG. 4: Free-energy difference between normal and superconducting states, experimentally obtained (open circles) and calculated from the four-band Eliashberg model (light gray line). The dashed, dash- dotted, and dotted lines represent the partial contributions of the in- dividual bands. Inset: Spin-fluctuation coupling function B (Ω) . e e r i m t o t a r a t u r e ( K h . 0 . 1 . 2 . 3 ( m namical spin susceptibility of Ba(Fe$_{1}$$_{−}$$_{x}$Co$_{x}$)$_{2}$As$_{2}$ [27, 28]. Here λ$_{ij}$ is the coupling constant for pairing of electrons in bands i and j . For our calculations we use the intraband cou- pling matrix elements λ$_{ii}$ = 0.2 in order to take account of the weak electron-phonon contribution [1], for interband re- pulsion λ$_{12}$ = λ$_{34}$ = 0 due to the symmetry of the wave functions, and λ$_{13}$ = λ$_{14}$ = − 1.0, λ$_{23}$ = λ$_{24}$ = − 0.2 ( λ$_{ji}$ = λ$_{ij}$N h,e i (0) /N h,e j (0) ). The corresponding densities of states are taken as N h $_{1}$(0) = 29 Ry − $^{1}$, N h $_{2}$(0) = 43 Ry − 1 for the hole bands, and as N e $_{3}$(0) = N e $_{4}$(0) = 8.5 Ry − 1 for the two equivalent electron bands [14]. The chosen param- eters, λ$_{ij}$ ( i = j ) and N h,e i (0) , allow the best simultaneous fit to the experimental values of T$_{c}$ , the superconducting gaps, and the temperature dependence of the free-energy difference, yielding T$_{c}$ = 38.5 K, and the following gap values: Δ h 1 = − 8.5 meV, Δ h 2 = − 3.6 meV for hole bands, and Δ e 3 = Δ e 4 = 9.2 meV for electronic ones. The effective coupling con- stant averaged over all bands, λ av = ∑ ij N h,e i (0) λ$_{ij}$/N$_{tot}$ , has a value of 1.9, remarkably close to the coupling constant reported for Pb$_{0}$$_{.}$$_{8}$Bi$_{0}$$_{.}$$_{2}$ [19] and in agreement with the con- clusions of our phenomenological analysis presented in Fig. 3. The consistency with the experiment tolerates some vari- ation of the parameters and remains satisfactory for $_{ℏ}$Ω max SF within the 10 - 20 meV energy range [14]. A shift of $_{ℏ}$Ω max SF to lower energy leads to a larger values of λ av with N h,e i (0) ap- proaching the results of density functional theory (DFT) [17]. This accounts for the strong renormalization of the Sommer- feld constant γ N $_{K}$/γ$_{DFT}$ ∼ 5 . The superconductivity-induced free-energy difference, Δ F ( T ) , was calculated by using the expressions in Ref. 21. 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