An experimental formula of the nucleation rate I of polyethylene as a function of number density of entanglement v, within the melt was obtained as I(nu(e)) proportional to exp(-gammanu(e)), where gamma is a constant. In order to obtain a functional form of I(nu(e)), I is determined by changing v, within the melt. The nu(e) within the melt can be changed when crystals with different lamellar thickness I are melted. It is shown that the nu(e) within the melt just after melting is related to I before melting. The v. of folded chain crystals (FCCs) is large, while that of extended chain single crystals (ECSCs) is very small. Therefore, strictly speaking, the experimental formula is a kind of 'semi-experimental' one. Because it is obtained by combining an experimental formula of I as a function of 1 before melting I(l) and a formula between I and v, based on the most probable model. It was found that the v, dependence of I is mainly controlled by the topological diffusion process within the interface between the melt and a nucleus and/or within the nucleus not by the forming process of a critical nucleus. The slope of the plots of log I against DeltaT(-2) was constant, irrespective of morphologies, FCCs and ECSCs, where AT is the degree of supercooling. From this fact, it was concluded that the fold type nucleus are formed from the melt of ECSCs as well as from the melt of FCCs. In our previous study, we found that I decreases exponentially with increase of annealing time At at a temperature above the melting temperature. From these results, we proposed a 'two-stage melt relaxation', i.e. fast conformational and slow topological relaxations. When the ECSCs are melted, extended chains within ECSCs are rapidly changed to random coiled chain conformation and then chains gradually entangle each other. We also proposed a formula, nu(e)(Deltat) proportional to -In{const. + A exp(-Deltat/tau(m))}, where A is a constant and tau(m) is the 'melt relaxation' time.