The Electronic Specific Heat of Ba$_{1-x}$K$_{x}$Fe$_{2}$As$_{2}$ from 2K to 380K

Using a unique differential technique, we have measured the specific heat capacity of polycrystalline Ba$_{1-x}$K$_{x}$Fe$_{2}$As$_{2}$ with x = 0, 0.1, 0.2, 0.3, 0.5 0.9 and 1.0, between 2K and 380K and in magnetic fields (H) from 0 -- 13T. We determine the electronic specific heat coefficient $\gamma $ ($\equiv $ C$_{el}$/T) over the entire range of T, H and x and compare it with the magnetic susceptibility of the seven samples. We show that our results are consistent with single crystal studies but give further interesting information. For x $<$ 0.3, $\gamma $ is progressively reduced at low T by a SDW gap, but is only weakly doping and T-dependent above the structural/magnetic transition. For x = 0.3 the normal state $\gamma _{n}$ is constant up to 380K, but as x increases from 0.3 to 1.0, $\gamma _{n}$ becomes increasingly T-dependent, increasing by a factor two at low-T and decreasing by a factor 1.5 at 380K for x = 1. We consider possible explanations for this striking T-dependence in terms of a sharp peak in the electronic density of states, a strongly x- and T-dependent effective mass enhancement, or low energy magnetic excitations. The H-dependent measurements allow us to extract the critical fields, superfluid density and coherence length as functions of doping and temperature.