  We use Grassmann algebra to study the phase transition in the two-dimensional
ferromagnetic Blume-Capel model from a fermionic point of view. This model
presents a phase diagram with a second order critical line which becomes first
order through a tricritical point, and was used to model the phase transition
in specific magnetic materials and liquid mixtures of He$^3$-He$^4$. In
particular, we are able to map the spin-1 system of the BC model onto an
effective fermionic action from which we obtain the exact mass of the theory,
the condition of vanishing mass defines the critical line. This effective
action is actually an extension of the free fermion Ising action with an
additional quartic interaction term. The effect of this term is merely to
render the excitation spectrum of the fermions unstable at the tricritical
point. The results are compared with recent numerical Monte-Carlo simulations.
