By studying the effect of thermodynamic and kinetic factors on the nucleation frequency of under-cooled metals, the expressions for heterogeneous and homogenous critical nucleation frequencies have been established. The results show that the homogenous critical nucleation frequency per unit volume is directly proportional to the ratio of the cooling rate to the volume of liquid metal. As the value of the equilibrium contact angle function f(θ) of the most effective catalyst is constant, the heterogeneous critical nucleation frequency per unit area of the catalyst surface is directly proportional to the ratio of the cooling rate to the sum of the surface area of the most effective catalyst surface ( Rc/ VSV) . When Rc/ VSC is constant, the heterogeneous critical nucleation frequency per unit area of the catalyst surface is inversely proportional to f(θ)0.53; the critical nucleation frequency per unit continuous mass of the metal melt for both the homogenous and heterogeneous nucleation can be expressed By studying the effect of thermodynamic and kinetic factors on the nucleation frequency of under-cooled metals, the expressions for heterogeneous and homogenous critical nucleation frequencies have been established. The results show that the homogenous critical nucleation frequency per unit volume is directly proportional to the ratio of the cooling rate to the volume of liquid metal. As the value of the equilibrium contact angle function f (θ) of the most effective catalyst is constant, the heterogeneous critical nucleation frequency per unit area of ​​the catalyst surface is directly proportional to the ratio of Rc / VSC is constant, the heterogeneous critical nucleation frequency per unit area of ​​the catalyst surface is inversely proportional to f (θ) 0.53; the critical nucleation frequency per unit continuous mass of the metal melt for both the homogenous and heterogeneous nucleatio n can be expressed