The Crab Nebula and the Pulsar have been pointed several times by the BeppoSAX satellite .
We analysed all these observations , summed together , to study the main spectral properties with a good phase resolution .
A new accurate estimate of the hydrogen column density in the Crab direction N _ { H } =3.23 \times 10 ^ { 21 } cm ^ { -2 } is given as derived from the analysis of the off-pulse emission at low energies .
We studied the changes of the spectral index with the pulse phase and showed that in the interpeak region it is systematically harder than in the main peaks of about 0.4 .
We observed also a significant spectral steepening of the pulsed emission and showed that this distribution can be represented by a unique law with a linear variable log slope and apply it to the spectrum of the first peak .
A two component model is proposed to explain the phase evolution of the spectrum .
One component has the same profile observed in the optical while the other presents a maximum at the phase 0.4 with essentially the spectrum of the interpeak region .
Two possible interpretations for the origin of the latter component are discussed .