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The phase diagram of the Heisenberg ferromagnetic model in the presence of amagnetic random field (we have used bimodal distribution) of spin S=1/2(quantum case) and $S=\infty $ (classical case) on a simple cubic lattice isstudied within the framework of the effective-field theory in finite cluster(we have chosen N=2 spins). Integrating out the part of order parameter(equation of state), we obtained an effective Landau expansion for the freeenergy written in terms of the order parameter $\Psi (m)$. Using Maxwellconstruction we have obtained the phase diagram in the $T-H$ plane for allinterval of field. The first-order transition temperature is calculated by thediscontinuity of the magnetization at $T_{c}^{\ast}(H)$, on the other hand inthe continuous transition the magnetization is null at $T=T_{c}(H)$. At nulltemperature (T=0) we have found the \textbf{coexistence} field $H_{c}=3.23J$that is independent of spin value. The transition temperature $T_{c}(H)$ forthe classical case ($S=\infty $), in the $T-H$ plane, is larger than thequantum case (S=1/2).

Study of the first-order phase transition in the classical and quantum random field Heisenberg model on a simple cubic lattice