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

where $d3w = v2dv sinhdhdf$ is the unit volume in phase space (see Figure 105

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.,

Since for a top-hat Equation 99 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

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