El rol de la actividad experimental en la construcción de la propuesta de enseñanza

In the literature, the kinetic energyEkinof the primary particle in the CEM is usually stated by the kinetic energy of the incident primary particle at CEM cone entrance (fig. 4.11). However, in many cases this particular definition is comparably vague and deviates significantly from the actual kinetic energy Ekin of the incident particle at the primary particle impact in the active surface of the CEM (stage g1; fig. 2.1). Especially for small kinetic energy values Ekin < 1 keV, the specific impact position of the primary particle in the CEM detector and therefore the correction of the impact energy of the incident particle due to the internal CEM

4.3. Charged particle optics -4.0 electric potential [kV] rel. z-distance [mm] -3.0 -2.0 -1.0 0.0 0 +10 +20 -10 -20 rel. z-distance [mm] electric potential [kV] -0.8 -0.6 -0.4 -0.2 0.0 -1.0 -4000 -3500 -3000 -2500 -2000 -1500 -1000 -500 0 -1000 -800 -600 -400 -200 0 150 -400 V -400 V -1000 V _{-400 V} -400 V -400 V (a) (b) (c) (d) 0 V 0 V 0 V 0 V ion CEM entrance ion CEM - e CEM - e CEM entrance ion CEM - e CEM linear approx. DUacc DUi ionisation center -4200 V -400 V 0 +10 +20 -10 -20 DUcor d 87 + Rb - e ion CEM entrance - e CEM entrance ionisation center ion. frag. flight axis 10 mm impact position DUe de di 0 -z +z

Figure 4.11: Particle accelerating potentials at the central flight axis of the photoionisation fragments in between the CEMs for two different potential differences∆Uacc. (a) Isoline plot as fig. 4.9 with a potential difference of ∆Uacc= 3800 Vbetween the two CEMs. (b) Illustration of the electric potential along the dashed line in (a), corresponding to the central flight axis of the photoionisation fragments into the respective CEM. (c) Isoline plot similar to (a), but with a potential difference of ∆Uacc= 600 Vin between the CEMs only. (d) Corresponding electric potential along the dashed line in (c).

gain potential has to be considered in more detail. In particular, for low kinetic energies (i.e., Ekin<1 keV) and a primary impact position deep in the CEM cone, the modification of the kinetic energyEkin due to the internal CEM potential will become macroscopic. This results as the internal gain field in the CEM becomes comparable to the external acceleration fields of the primary particles up to the CEM cone (fig. 4.11(d)). For comparison to the literature, the knowledge of the exact impact energy is especially important in relation to observed isolated values of the primary particle emission yieldδ0, which will likewise affect the measured CEM detector quantum yield valuesηdetector (see section 2.3). At low kinetic energies at the CEM cone entrance, the influence of the spatial position on the actual kinetic energy Ekin of the incident particle at impact and therefore on the detection efficiency of the particular CEM detector can be observed in the measurements of, e.g., [87, 88, 101]. In the following, the individual energy correction for incident 87Rb-ions and electrons in the joint CEM detection system is thus investigated.

Correction of the kinetic impact energy due to the internal CEM field

Figure 4.11 shows the influence of the respective internal CEM field on both photoionisation fragments, individually. In both cases, the influence of the internal gain field explicitly depends on the spatial impact position of the primary particle in the CEM. Specifically, it is directly proportional to the distancedi (de for the electron) the particle has to travel from the cone entrance until primary particle impact at the active CEM surface. Noteably, the particular distances in the CEM will be in the order ofdi,e≈10 mmfor an impact position at the CEM channel wall (fig. 4.11(b,d), atz=∓18 mm) in contrast to an average cone hit with a distance of di,e ≈ 4.5 mm (z ≈ ∓12.5 mm). The individual potential difference in the corresponding CEM from the cone entrance up to the impact position of the primary particle is defined as

∆Ui,e=Ei,e· di,e. (4.1)

Here, the electric field in the CEM is defined as Ei,e = ∓UCEM/l, where UCEM represents the individual CEM gain detector voltage (see section 3.3), and l is the entire CEM channel length (see appendix A.3). Note that for incident primary ions, the internal CEM field up to primary particle impact is repulsive, for primary electrons it is further accelerating (see subsection 4.3.3).

In the joint CEM detection system, the actual impact energy Ekin at primary particle hit will remain comparably unaffected for large acceleration voltages ∆Uacc in between the CEMs (fig. 4.11(b)). However, for small acceleration voltages ∆Uacc < 1 kV especially the 87_{Rb-ion is significantly affected. Eventually, it will not even have sufficient kinetic energy} for a primary particle impact at the CEM surface at the CEM channel wall. As illustrated in fig. 4.11(d), the 87Rb-ion will therefore be repelled before reaching the active secondary emitting surface due to the repulsive internal potential in the ion-CEM(−∆Ui). In contrast to that, incident electrons in thee−-CEM are even further accelerated by the internal potential

(+∆Ue). Correspondingly, the kinetic energiesEkin of the primary particles (87Rb-ion,e−) at primary particle impact are individually corrected in accordance with the linear time-of-flight model of subsection 4.4 by

Ekin(di,e) =e0(∆Uacc/2∓∆Ui,e). (4.2) At low kinetic energies at the CEM cone entrance (Ekin <1 keV), this leads to a significant modification of the actual kinetic energy Ekin(di,e) at primary particle impact compared to

4.3. Charged particle optics 0 -1.0 -2.0 -3.0 -4.0 DU [kV]acc 0 1.0 2.0 3.0 4.0 DUcor electric potential [kV]

Figure 4.12: Electric potential of the ion-CEM entrance (black), the position of the ionisa- tion center at d/2 between the CEMs (blue), and the e−-CEM entrance (red) as illustrated in fig. 4.11(a). The dashed lines indicate the potentials corresponding to the constant approxi- mation of the electric field along the central flight axis. The scattered points show the electric potentials derived from similar simulations as displayed in fig. 4.11.

the kinetic energy at cone entrance as frequently stated in the literature. For the current setup of this thesis, the latter value would only correspond toEkin ≈∆Uacce0/2, in contrast to the expression of eq. 4.2. Note that for the modified relative potentials of the refined time-of-flight model, the ion flight distancedi in the ion-CEM has to be substituted todiCEM=di−dsaddle (fig. 4.13(b)).

Influence of the compensation electrodes

In a more refined version of the potential simulations, the linear scaling of the accelerating potential∆Uacc in fig. 4.11 is only an approximation. In this approximation, each of the two photoionisation fragments being ionised at d/2 experiences an identical accelerating poten- tial difference of ∆Uacc/2 until CEM cone entrance. However, in contrast to the constant approximation of the electric field between the two CEMs, the particular electric potential of the compensation electrodes modifies the accelerating potential gradient between the CEMs. In fig. 4.11, this particular situation is illustrated when the electrodes are held at the same potential as the cone entrance aperture of thee−-CEM.

Figure 4.12 shows the deviation of the electric potential from the linear approximation at three selected positions along the central flight axis (red lines; fig. 4.11(a)). The three positions are the electric potential at the ion-CEM entrance, the potential of the ionisation center atd/2between the CEMs, and of thee−-CEM entrance as illustrated by the solid lines in fig. 4.11(a,b). From the electric potentials at the two CEM entrance positions, the relative accelerating potential difference of∆Uacc between the two CEMs is derived. In fig. 4.12, the dashed lines show the electric potential corresponding to the linear approximation according to the time-of-flight model introduced in section 4.4. The scattered points illustrate the electric potential as derived from the simulation displayed in fig. 4.11, respectively. At the ionisation regionz=d/2, the center potential is significantly shifted to the cone potential of thee−-CEM (∆Ucor) compared to the linear approximation, resulting in a relatively smaller potential difference up to thee−-CEM and a larger difference up to the ion-CEM. As a result, instead of ∆Uacc/2 each, the 87Rb-ion experiences 72 % of the overall accelerating potential difference ∆Uacc and the electron only 28 %, correspondingly. This discrepancy will affect

the time-of-flight model of subsection 5.4, altering the individual flight times (ti,te) of both generated photoionisation fragments starting at d/2.

Corrections of the kinetic energy at CEM cone entrance due to particle impact position

For the joint CEM detection system of this thesis, the actual kinetic energy Ekin at primary particle impact can be determined from the estimated impact position of the incident particle in the corresponding CEM detector. Using the modified potentials of the refined model with

∆Umod,i = 0.715·∆Uacc/2 and ∆Umod,e = 0.285·∆Uacc/2 as obtained from the potential simulations (fig. 4.12), the correction of the kinetic energies at particle impact according to the applied acceleration voltage difference ∆Uacc between the CEMs is

Ekin,i(∆Uacc, Ei, diCEM) = 0.715·∆Uacce0/2−e0EidiCEM, (4.3) for the ion-CEM, and

Ekin,e(∆Uacc, Ee, de) = 0.285·∆Uacce0/2 +e0Eede, (4.4) for the e−-CEM, accordingly. In the previous equations, the covered path length diCEM and de of the primary particles in the CEMs are obtained form the observed flight timesti andte of the photoionisation fragments (see section 5.3), in accordance with the flight time model of section 4.4. Correspondingly, the parametersdiCEM and de in turn determine the kinetic energy correction by the internal CEM potential∆Ui,e.