A quantum field of $s=1/2$ in the vicinity of a dust shell contracting at a distance $r({t}_{0})$ to near its gravitational radius ${r}_{s}$ as seen by a FIDO observer is considered. Such an observer perceives a batch of particles around the event horizon. The origin of the particles around the spherical surface of radius $r={r}_{s}+\ensuremath{\epsilon}$ lies in the thermal excitations in the Boulware vacuum state, ${|0⟩}_{B}$ for an external observer. The foregoing is done based on thermo field dynamics, as it allows one to explain the origin of ${S}_{\mathrm{BH}}$ as a state of entanglement between the modes of the fermionic field spreading through the Kruskal variety ${S}_{\mathrm{Ent}}\ensuremath{\propto}{S}_{\mathrm{BH}}$ with respect to a FIDO observer. A location of the degrees of freedom responsible for ${S}_{\mathrm{BH}}$ entropy is given. The occupation number for particles of a half-integer spin $s=1/2$ is estimated, and it is compared with the occupation number of particles of spin $s=0$, finding that the occupation number of the Dirac field is slightly lower than the occupation number of scalar particles in the vicinity of the gravitational radius; this allows confirming the exclusion principle near event horizons. The other thermodynamic properties of the field are estimated.