3 l n / T I n ( 0 . 8 P 0 . 8 0 . 2 ( = 1 . 8 ) 0 . 6 8 0 . 3 2 FIG. 3: Comparison of the dimensionless superconducting jump, en- ergy gap at T = 0 , and slope of the specific heat near T$_{c}$ calculated by the semiphenomenological expressions in Ref. 19 (lines) with our results for Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ (open circles), and for Pb$_{0}$$_{.}$$_{8}$Bi$_{0}$$_{.}$$_{2}$ (triangles) and In (squares), after Ref. 19. The open and solid circles represent the maximum gap values obtained by the two-band α and four-band Eliashberg model, respectively. Shaded areas represent the uncertainty in ω$_{ln}$/T$_{c}$ according to the error bars in our data points. same Δ C$_{el}$ ( T ) /T (within the symbol size in Fig. 2) has also been obtained by representing the lattice contribution on a ba- sis of Einstein modes [14]. The value of the Sommerfeld con- stant γ N $_{K}$= 50 mJ/mol K 2 is in between those reported for the pristine end compounds BaFe$_{2}$As$_{2}$ (6.1 mJ/molK 2 [15]) and KFe$_{2}$As$_{2}$ (69.1 mJ/molK 2 [16]). This value exceeds the one calculated from the band structure ( γ$_{DFT}$ ≈ 10.1 mJ/molK 2 for Ba$_{0}$$_{.}$$_{6}$K$_{0}$$_{.}$$_{4}$Fe$_{2}$As$_{2}$ [17]) by almost a factor of five, while it is consistently inferred from all specific heat experiments reported to date [3, 4, 11]. The thermodynamic critical field, H$_{c}$ ( T ) , can be obtained from the free-energy difference between the normal and su- perconducting states: Δ F ( T ) = µ$_{0}$V$_{m}$H 2 $_{c}$( T ) / 2 = Δ U ( T ) − T Δ S ( T ) (2) where V$_{m}$ = 6 . 05 × 10 − 5 m $^{3}$/mol is the molar volume determined from our X-ray diffraction measurements on Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ . The temperature dependence of the ther- modynamic critical field obtained by numerical integration of our data is displayed in the inset of Fig. 2. By fit- ting the thermodynamic critical field expressed as H$_{c}$ ( T ) = H$_{c}$$_{0}$ $^{(}$1 − ( T/T$_{c}$ ) 2 $^{)}$to our data between 2 and 38.5 K we ob- tained µ$_{0}$H$_{c}$$_{0}$ = 0.85 T, a value comparable to those of the cuprate superconductors [18]. In contrast to γ N $_{K}$, the superconductivity-induced elec- tronic specific heat is very sensitive to the sample qual- ity and phase purity. We now show that if the impu- rity scattering is minimized, Δ C$_{el}$ ( T ) reveals the intrin- sic multigap strong-coupling nature of the superconduct- ing state. Two quantities are commonly used to assess the pairing strength, namely the reduced jump anomaly, Δ C$_{el}$ ( T$_{c}$ ) / ( γ N $_{K}$T$_{c}$ ) , and the normalized slope of Δ C$_{el}$ ( T ) right below T$_{c}$ , g ( T$_{c}$ ) = − ∂ [Δ C$_{el}$ ( T ) / ( γ N $_{K}$T$_{c}$ )] /∂T |$_{T}$$_{c}$ . Their magnitudes in Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ , 2.5 and 11.1, re- spectively, are significantly larger than the corresponding BCS values of 1.43 and 3.77. In the framework of the Eliash- berg formalism, the coupling strength can be expressed in terms of the ratio $_{ℏ}$ω$_{ln}$ /k$_{B}$T$_{c}$ , where ω$_{ln}$ is the logarithmi- cally averaged frequency of bosons mediating the Cooper pairing [19]. In Figure 3, we place the reduced superconduct- ing jump and g -value of Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ on a universal plot [19] as a function of this dimensionless coupling param- eter. Both values correspond to $_{ℏ}$ω$_{ln}$ /k$_{B}$T$_{c}$ = 7 . 5 , close to the prototypical strong-coupling superconductor Pb$_{0}$$_{.}$$_{8}$Bi$_{0}$$_{.}$$_{2}$. This comparison indicates strong-coupling superconductivity in Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ . Finally, we illustrate that the strong coupling scenario can be described by fully microscopic calculations in the frame- work of a spin-fluctuation model. Such models generically yield s -wave superconducting gaps with different signs on dif- ferent bands (“ s$_{±}$ state”) [22]. It was shown in Refs. 23, 24 that strong-coupling effects lead to merging of the gaps in a simple two-band s$_{±}$ model. Therefore, a more realistic four- band model based on a band-structure with two hole bands and two electron bands crossing the Fermi surface needs to be considered [25]. The main input into the Eliashberg equa- tions is the spectral function of the intermediate bosons. Fol- lowing Ref. 26 we took a spin-fluctuation coupling func- tion ˜ B$_{ij}$ (Ω) = λ$_{ij}$B (Ω) [inset of Fig. 4] with a linear Ω - dependence at low frequencies, a maximum at $_{ℏ}$Ω max SF = 18 meV, and a fast decay at Ω > $_{ℏ}$Ω max SF , in qualitative agree- ment with recent experimental data on the normal-state dy- defined according to Δ$_{j}$( T ) = ( α$_{j}$ / α$_{BCS}$ ) Δ$_{BCS}$ , where α$_{BCS}$ = Δ$_{BCS}$(0) /k$_{B}$T$_{c}$ = 1 . 764 is the weak-coupling value of the gap ratio. Another set of adjustable parameters is the fractions of the total electronic density of states, γ$_{j}$/γ N $_{K}$, that each band contributes to the superconducting condensate. Fig- ure 2 shows the result of our multigap fit (black solid curve) with only two gaps survived α$_{1}$ = 3.3, α$_{2}$ = 1.1, and γ$_{1}$ ∼ γ$_{2}$ ∼ 0 . 5 · γ N $_{K}$, which reproduces well Δ C$_{el}$ ( T ) /T below T$_{c}$ . One of the gaps, Δ$_{2}$(0) = 3.5 meV, is somewhat smaller than Δ$_{BCS}$(0) , whereas the other one, Δ$_{1}$(0) = 11 meV, is much larger. Our analysis thus yields thermodynamic evidence of two different superconducting gaps (or groups of gaps) with quite different absolute values in Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ , in agreement with ARPES and µ SR results [5]. Further, mark- ing the corresponding point, 2Δ $^{max}$/k$_{B}$ T$_{c}$ = 6.6, on the uni- versal plot of Fig. 3 strongly supports our conclusion about the strong-coupling nature of superconductivity in this com- pound. The same issue can be addressed by considering the mag- nitude of the superconducting energy gap. Clearly, the promi- nent knee in the Δ C$_{el}$ ( T ) /T data around T = 15 K (Fig. 2) cannot be described in terms of models with a single gap. We have hence fitted these data to the phenomenological multi- band α -model [20], which assumes a BCS temperature de- pendence of the gaps and has been widely used to analyze heat capacity data on MgB$_{2}$ [21]. The superconducting gap mag- nitudes at T = 0 are introduced as adjustable parameters α$_{j}$ ,