GLYPH<s48> GLYPH<s49>GLYPH<s48> GLYPH<s50>GLYPH<s48> GLYPH<s51>GLYPH<s48> GLYPH<s52>GLYPH<s48> GLYPH<s48> GLYPH<s53> GLYPH<s49>GLYPH<s48> GLYPH<s49>GLYPH<s53> GLYPH<s50>GLYPH<s48> GLYPH<s48> GLYPH<s49> GLYPH<s50> GLYPH<s115>GLYPH<s32>GLYPH<s43>GLYPH<s32> GLYPH<s105> GLYPH<s100> GLYPH<s32>GLYPH<s119>GLYPH<s97>GLYPH<s118>GLYPH<s101> GLYPH<s115> GLYPH<s32>GLYPH<s119>GLYPH<s97>GLYPH<s118>GLYPH<s101> GLYPH<s100> GLYPH<s32>GLYPH<s119>GLYPH<s97>GLYPH<s118>GLYPH<s101> Δ GLYPH<s32>GLYPH<s40>GLYPH<s109>GLYPH<s101>GLYPH<s86>GLYPH<s41> GLYPH<s84> GLYPH<s101>GLYPH<s109>GLYPH<s112>GLYPH<s101>GLYPH<s114>GLYPH<s97>GLYPH<s116>GLYPH<s117>GLYPH<s114>GLYPH<s101>GLYPH<s32>GLYPH<s40>GLYPH<s75>GLYPH<s41> GLYPH<s68>GLYPH<s101>GLYPH<s110>GLYPH<s115>GLYPH<s105>GLYPH<s116>GLYPH<s121>GLYPH<s32>GLYPH<s111>GLYPH<s102> GLYPH<s32>GLYPH<s83>GLYPH<s116>GLYPH<s97>GLYPH<s116>GLYPH<s101>GLYPH<s115> GLYPH<s32>GLYPH<s69>GLYPH<s47> Δ GLYPH<s48> GLYPH<s50> GLYPH<s48> GLYPH<s52> GLYPH<s48> GLYPH<s54> GLYPH<s48> GLYPH<s56> GLYPH<s48> GLYPH<s49> GLYPH<s48> GLYPH<s48> GLYPH<s50> GLYPH<s53> GLYPH<s53> GLYPH<s48> GLYPH<s55> GLYPH<s53> GLYPH<s49> GLYPH<s48> GLYPH<s49> GLYPH<s50> GLYPH<s53> GLYPH<s49> GLYPH<s53> GLYPH<s48> GLYPH<s48> GLYPH<s53> GLYPH<s48> GLYPH<s49> GLYPH<s48> GLYPH<s48> GLYPH<s49> GLYPH<s53> GLYPH<s48> GLYPH<s50> GLYPH<s48> GLYPH<s48> GLYPH<s50> GLYPH<s53> GLYPH<s48> GLYPH<s51> GLYPH<s48> GLYPH<s48> GLYPH<s51> GLYPH<s53> GLYPH<s48> GLYPH<s48> GLYPH<s53> GLYPH<s48> GLYPH<s49> GLYPH<s48> GLYPH<s48> GLYPH<s45>GLYPH<s50> GLYPH<s53> GLYPH<s48> GLYPH<s50> GLYPH<s53> GLYPH<s53> GLYPH<s48> GLYPH<s55> GLYPH<s53> GLYPH<s48> GLYPH<s49> GLYPH<s48> GLYPH<s48> GLYPH<s50> GLYPH<s48> GLYPH<s48> GLYPH<s51> GLYPH<s48> GLYPH<s48> GLYPH<s48> GLYPH<s54> GLYPH<s49> GLYPH<s50> GLYPH<s32>GLYPH<s120> GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s48>GLYPH<s46>GLYPH<s51> GLYPH<s32>GLYPH<s120> GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s48>GLYPH<s46>GLYPH<s49> GLYPH<s32>GLYPH<s120> GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s48> γ GLYPH<s32>GLYPH<s40>GLYPH<s109> GLYPH<s74> GLYPH<s32>GLYPH<s109> GLYPH<s111>GLYPH<s108> GLYPH<s45>GLYPH<s49> GLYPH<s32>GLYPH<s75> GLYPH<s45>GLYPH<s50> GLYPH<s41> GLYPH<s84>GLYPH<s101>GLYPH<s109> GLYPH<s112>GLYPH<s101>GLYPH<s114>GLYPH<s97>GLYPH<s116>GLYPH<s117>GLYPH<s114>GLYPH<s101>GLYPH<s32>GLYPH<s40>GLYPH<s75>GLYPH<s41> GLYPH<s32> GLYPH<s32>GLYPH<s48>GLYPH<s46>GLYPH<s51>GLYPH<s32>GLYPH<s45>GLYPH<s32>GLYPH<s48>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s48>GLYPH<s84> GLYPH<s32>GLYPH<s48>GLYPH<s46>GLYPH<s51>GLYPH<s32>GLYPH<s45>GLYPH<s32>GLYPH<s48>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s49>GLYPH<s51>GLYPH<s84> GLYPH<s32>GLYPH<s48>GLYPH<s46>GLYPH<s49>GLYPH<s32>GLYPH<s45>GLYPH<s32>GLYPH<s48>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s48>GLYPH<s84> GLYPH<s40>GLYPH<s98>GLYPH<s41> GLYPH<s40>GLYPH<s97>GLYPH<s41> Δγ GLYPH<s32>GLYPH<s116>GLYPH<s111>GLYPH<s116> GLYPH<s32>GLYPH<s40>GLYPH<s109> GLYPH<s74> GLYPH<s32>GLYPH<s109> GLYPH<s111>GLYPH<s108> GLYPH<s45>GLYPH<s49> GLYPH<s32>GLYPH<s75> GLYPH<s50> GLYPH<s41> GLYPH<s32> γ GLYPH<s32>GLYPH<s97>GLYPH<s110> GLYPH<s32> FIG. 3: (Color online) Temperature dependence of the super- conducting gap extracted by comparing the experimentally determined S/T with that calculated from models of the den- sity of states (shown in the inset) given by Eqns. 1-3. FIG. 2: (Color online) (a) The directly measured raw differ- ence in specific heats between the doped samples and undoped reference in 0 and 13T fields. The inset shows the anhar- monic phonon term γ $^{an}$. (b) The electronic specific heat of Ba$_{1}$$_{−}$$_{x}$K$_{x}$Fe$_{2}$As$_{2}$ for x = 0, 0.1 and 0.3. N ( E ) = E/ $^{√}$E 2 − Δ 2 (1) N ( E ) = 1 N$_{θ}$ ∑ θ E/ $^{√}$E 2 − Δ $^{2}$cos$^{2}$2 θ (2) N ( E ) = 1 N$_{θ}$ ∑ θ E/ $^{√}$E 2 − 0 . 98Δ $^{2}$cos$^{2}$2 θ − 0 . 02Δ 2 (3) 3 γ $^{ph}$( x = 0 . 3) gives γ $^{ph}$( x = 0) and γ $^{ph}$( x = 0 . 1). The electronic specific heats of the three samples are shown in Figure 2(b). Above 150K, γ is at most only weakly T -dependent and almost independent of doping. At low temperatures γ for the x = 0 and 0.1 samples is heavily reduced by factors of ∼ 11 and 5 respectively due to the opening of a spin density wave (SDW) gap be- low T$_{0}$ ∼ 136K, and increases as ∼ 20 × 10 − $^{6}$T 3 mJ/mol K 2 in the range 40 − 110K. Band splitting and signs of partial gapping of the Fermi surface have been observed in the SDW state of BaFe$_{2}$As$_{2}$ by angle-resolved photoe- mission spectroscopy[16]. γ for the x = 0.3 sample is dominated by the large superconducting anomaly with Δ γ ( T$_{c}$ ) ≈ 48 mJ/mol K $^{2}$, a normal-state γ$_{N}$ (0) of about 47 mJ/mol K 2 and γ (0) = 1.8 mJ/mol K 2 . Measure- ments on a more overdoped crystal with x = 0.4 by Mu et al .[17] show an even larger Δ γ ( T$_{c}$ ) ≈ 100 mJ/mol K 2 and γ$_{N}$ (0) ≈ 63 mJ/mol K $^{2}$, implying a further growth in the density of states with doping. For our x =0.3 sample the superconducting condensation energy U (0) = 10.6 J/mol, as determined from ∫ T$_{c}$ 0 ( S$_{n}$ − S$_{s}$ ) dT , where S$_{n}$ and S$_{s}$ are the normal and superconducting state en- tropies respectively. In Fig. 2(b), the curvature in γ for x =0.3 between 10 and 20K is reduced compared to the raw Δ γ tot data (Fig. 2(a)) by the phonon correction de- scribed above. Evidence that the remaining curvature in γ may reflect a genuine anomalous T -dependence of the electronic term comes from the observation of similar anomalous curvature between 15 and 20K in the field de- pendence Δ γ ( H ) = γ ( H, T ) − γ (0 , T ) shown in the inset to Fig. 4(a). Since the phonon term is H -independent, this anomalous curvature in Δ γ ( H ) can only be of elec- tronic origin, and may signal the presence of a second energy gap as inferred by Mu et al [17]. To determine the magnitude and temperature depen- dence of the superconducting gap Δ for x = 0.3 (shown in Fig. 3) we first subtract from γ the small residual γ (0) = 1.8 mJ/mol K 2 and then match the corrected S/T with that calculated assuming the following models for the density of states (see Fig. 3 inset): s -wave d -wave and s + id -wave where the summations over θ are from 0 to 45 ◦ and N$_{θ}$ is the number of θ values. Although such modelling is too crude to pin down the exact nature of the gap, the rapid increase in Δ below 15K in the case of the as- sumed d -wave gap function, shows that nodes on the Fermi surface are incompatible with the T -dependence of γ observed at low temperature, even in the presence of weak pair breaking. The s + id -wave gap function pro- duces a more reasonable Δ( T ). This function completely gaps the Fermi surface but is strongly anisotropic. Such anisotropy could arise from the presence of multiple gaps