  The two-gap structure in the superconducting state of MgB_2 gives rise to
unusual thermodynamic properties which depart markedly from the isotropic
single-band BCS model, both in their temperature- and field dependence. We
report and discuss measurements of the specific heat up to 16 T on ceramic, and
up to 14 T on single crystal samples, which demonstrate these effects in the
bulk. The behavior in zero field is described in terms of two characteristic
temperatures, a crossover temperature Tc_pi ~ 13 K, and a critical temperature
Tc = Tc_sigma ~ 38 K, whereas the mixed-state specific heat requires three
characteristic fields, an isotropic crossover field Hc2_pi ~ 0.35 T, and an
anisotropic upper critical field with extreme values Hc2_sigma_c ~ 3.5 T and
Hc2_sigma_ab ~ 19 T, where the indexes \pi and \sigma refer to the 3D and 2D
sheets of the Fermi surface. Irradiation-induced interband scattering tends to
move the gaps toward a common value, and increases the upper critical field up
to ~ 28 T when Tc = 30 K.
