Definition of scattered particle?

Question

Compare the number of scattered particles:
$N_s=Fa\int\sigma(\theta)d\Omega$
With the total number of incident particles:
$N_{in}=Fa$

Here, F is the flux of incoming beam, a the area. sigma the crossection and omega the solid angle.

Why isnt $N_s=N_{in}$? How does one define which particles are scattered and which are not, arent they all interacting with the target to some degree? Isnt particles conserved normally?

Do almost all the particles either pass right through almost undetected or are scattered signficiantly, so what we are really integrating over is a sphere surrounding the target except a spot of area $a$ where the beam exits?

Answer 1

With the exception of stopping targets, there is going to be a portion of the beam which continues down the beam pipe (i.e. misses even the innermost elements of the detector), which would usually be counted as "unscattered" for that purpose.

In that case the integration is often over the solid angle covered by active detector elements.