The application of the muon-spin rotation/relaxation ($mu$SR) technique for studying type-I superconductivity is discussed. In the intermediate state, i.e. when a type-I superconducting sample with non-zero demagnetization factor $N$ is separated into normal state and Meissner state (superconducting) domains, the $mu$SR technique allows to determine with very high precision the value of the thermodynamic critical field $B_{rm c}$, as well as the volume of the sample in the normal and the superconducting state. Due to the microscopic nature of $mu$SR technique, the $B_{rm c}$ values are determined directly via measurements of the internal field inside the normal state domains. No assumptions or introduction of any type of measurement criteria are needed. Experiments performed on a classical type-I superconductor, a cylindrically shaped $beta-$Sn sample, allowed to reconstruct the full $B-T$ phase diagram. The zero-temperature value of the thermodynamic critical field $B_{rm c}(0)=30.578(6)$ mT and the transition temperature $T_{rm c}=3.717(3)$ K were determined and found to be in a good agreement with the literature data. An experimentally obtained demagnetization factor is in very good agreement with theoretical calculations of the demagnetization factor of a finite cylinder. The analysis of $B_{rm c}(T)$ dependence within the framework of the phenomenological $alpha-$model allow to obtain the value of the superconducting energy gap $Delta=0.59(1)$ meV, of the electronic specific heat $gamma_e=1.781(3)$ ${rm mJ}/{rm mol}; {rm K}^2$ and of the jump in the heat capacity ${Delta C(T_c)}/{gamma T_{rm c}}=1.55(2)$.
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