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16.2 Measuring the velocity distribution function

In this section, we will show how to reconstruct the average density of the distribution function starting from the particles detected by the analyzer. Let us consider the flux through a unitary surface of particles coming from a given direction. If f (v ,v ,v ) x y z is the particle distribution function in phase space, f (vx, vy,vz)dvxdvydvz is the number of particles per unit volume 3 (pp/cm ) with velocity between vx and vx + dvx, vy and vy + dvy,vz and vz + dvz, the consequent incident flux Pi through the unit surface is
integral integral integral Pi = vfd3w, (100)
where d3w = v2dv sinhdhdf is the unit volume in phase space (see Figure 105View Image).
View Image

Figure 105: Unit volume in phase space.
The transmitted flux Ct will be less than the incident flux P i because not all the incident particles will be transmitted and Pi will be multiplied by the effective surface S(< 1), i.e.,
integral integral integral integral integral integral Ct = Svf d3w = Svf v2dv sin hdhdf (101)

Since for a top-hat Equation 99View Equation is valid, then

2 3dv- 3 v dv = v v ~ v .

We have that the counts recorded within the unit phase space volume would be given by

Ct = f Sv4dhdf dv-sinh = f v4G, (102) f,h,v f,h,v v f,h,v
where G is called Geometrical Factor and is a characteristic of the instrument. Then, from the previous expression it follows that the phase space density function ff,h,v can be directly reconstructed from the counts
Ctf,h,v- ff,h,v = v4G . (103)

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