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The superconducting state of Mg$^{10}$B$_{2}$ is investigated by specificheat measurements in detail. The specific heat in the normal state is analyzedusing a recently developed computer code. This allows for an extraction of theelectronic specific heat in the superconducting state with high accuracy and afair determination of the main lattice features. One of the two investigatedsamples shows a hump in the specific heat at low temperatures within thesuperconducting state, accompanied by an unusual low value of the small gap,$\Delta_{\pi}(0)=1.32 meV$, pointing to a very weak inter-band coupling. Thissample allows for a detailed analysis of the contribution from the $\pi$-bandto the electronic specific heat in the superconducting state. Therefore theusual analysis method is modified, to include the individual conservation ofentropy of both bands. From analyzing the deviation function $D(t)$ ofMgB$_{2}$, the theoretically predicted weak inter-band coupling scenario isconfirmed.Comment: major revision

physical review. b, condensed matter and materials physics

Weak interband coupling in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mi>Mg</mml:mi><mml:mn>10</mml:mn></mml:msup><mml:msub><mml:mi mathvariant="normal">B</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math>: A specific-heat analysis

Weak interband coupling inMg10B2: A specific-heat analysis