Appendix A: Pile-up probability for split events

  To avoid lengthy equations I denote as p₁ and p₂ the probability to produce a monopixel event (from one or more monopixels) and a vertical (or horizontal) bipixel event (from one or more vertical or horizontal bipixels).

$$\begin{eqnarray} p_1 & = & 1 \: - \: \mathrm e^{- \alpha_1 \lambda} \\ p_2 & = & 1 \: - \: \mathrm e^{- \alpha_2 \lambda / 2}.\end{eqnarray}$$ (A1) (A2)

A.1 Diagonal and extended monopixels

It is interesting to derive the rate of diagonally attached monopixels (two monopixels touching by a corner and globally isolated. See Fig. 1, top centre). Those are produced only by piled-up monopixels (and Si$_{\rm K}$ escape, which can be recognized because one of the energies is that of Si$_{\rm K}$). Therefore they can reliably be counted as two monopixels. The exclusion area corresponding to clean (not piled-up) diagonal events, derived in the same way as in 2.2, is 14 for monopixels, 18 for bipixels, 21 or 22 for tripixels (depending on their orientation), and 23 for quadripixels. The production probability is the square of that of monopixels, times 4 (there are 4 corners), both for $\mu_1^{\rm d}$ (total) and for $\mu_1^{\rm d \, t}$ (clean). Note that the rate of diagonal events (comprising two monopixels) is half that.

$$\begin{eqnarray} \gamma_1^{\rm d} & = & 14\: + \: 4\,\alpha_2 \: + \: 7.5\,\alph... ...: (\alpha_1 \lambda)^2 \; \mathrm e^{-\gamma_1^{\rm d} \lambda}.\end{eqnarray}$$ (A3) (A4)

For completeness I note that if one relaxes the constraint that a monopixel be fully isolated, so as to include monopixels touching anything else by a corner (Fig. 1, top right), only $\gamma_1$ is changed in Eqs. (2-3) to become  

$$\gamma_1^{\rm c} \; =\; 5\: + \: 3\,\alpha_2 \: + \: 5\,\alpha_3 \: + \: 7\,\alpha_4 .$$ (A5)

These events are less reliable, though, because they can also be produced by cosmic-rays.

A.2 Bipixels

For clean bipixels (Fig. 1, bottom left) the exclusion areas are [12, 15.5, 19, 20]. 15.5 for bipixels is actually 15 for bipixels parallel to that considered and 16 for perpendicular bipixels. For $\mu_2$ (all bipixels) one must remove the two pixels of the pattern for monopixels, and one pixel for parallel bipixels, from the exclusion areas. The probability to produce a bipixel pattern of a given orientation (vertical, say), is that of having at least one vertical bipixel there (p₂) plus in the remaining cases (1-p₂) that of having at least two adjacent monopixels (p₁²).

$$\begin{eqnarray} \gamma_2 & = & 12\:+\:3.5\,\alpha_2\:+\:7\,\alpha_3\:+\:8\,\alp... ... t} & = & (\alpha_2 \lambda) \; \mathrm e^{- \gamma_2 \lambda} .\end{eqnarray}$$ (A6) (A7)

A.3 Tripixels

  For clean tripixels (Fig. 1, bottom center) the exclusion areas are [15, 19, 22.75, 24]. 22.75 is due to the fact that the exclusion area is not the same for tripixels having exactly reversed orientation as that considered (22) and all others (23). For $\mu_3$ (all tripixels) one must remove the three pixels of the pattern for monopixels, one pixel for bipixels of all orientations and one pixel for identical tripixels, from the exclusion areas.

For tripixels there is a complication since the easiest thing to compute is the probability $\mu_3^{\rm g}$ to get a geometrical tripixel, but what one is really interested in is the probability $\mu_3$ to get a tripixel with maximum charge at the corner (other tripixels cannot be confused with a regular X-ray and can be rejected straight away). $T_3^{\rm g}$ (and T₃) is the probability to construct a tripixel from mono- and bipixels. It is written as the sum of that with no bipixel (requiring three monopixels), plus that with one (or more) bipixel of a single orientation (requiring one additional monopixel), plus that with bipixels of both orientations. I obtain an estimate of T₃ by noting that a) exactly one third of tripixels from three monopixels have maximum energy at the corner; b) for tripixels formed from one bipixel plus one monopixel only half the bipixels have their maximum charge at the corner, and that charge has at most a 50% chance of being larger than that of the monopixel; and c) for tripixels formed from two bipixels only the case when none of the bipixels has maximum charge at the corner is excluded. The last two contributions to T₃ (b and c) can only be estimated approximately.

$$\begin{eqnarray} \gamma_3 & = & 15\:+\: 4\,\alpha_2 \: + \: 7.75\,\alpha_3 \: + ... ...m t} & = & (\alpha_3 \lambda) \; \mathrm e^{-\gamma_3 \lambda} .\end{eqnarray}$$ (A8) (A9)

A.4 Quadripixels

For clean quadripixels (Fig. 1, bottom right) the exclusion areas are [16, 20, 24, 25]. For $\mu_4$ (all quadripixels) one must remove the four pixels of the pattern for monopixels, two pixels for bipixels of any orientation and one pixel for tri- and quadripixels, from the exclusion areas. Q₃ below is the probability to get a quadripixel event from mono-, bi- and tripixels. Q₂ is the probability to get a quadripixel event from mono- and bipixels only. In Q₃ the second term (subtracted) is the probability to have no tripixel and not the right combination of mono- or bipixels. The rightmost term (also subtracted) is the probability to have one (or more) tripixel of a single orientation and no bipixel or monopixel filling the hole. All other combinations form a quadripixel. Q₂ is written as the sum of the probability to get a quadripixel with no bipixel (requiring four monopixels), plus that with a bipixel at a single place in a single orientation (requiring two additional monopixels), plus that with bipixels of both orientations but no two parallel bipixels (requiring one additional monopixel), plus that with two parallel bipixels.

$$\begin{eqnarray} \gamma_4 & = & 16\:+\: 4\,\alpha_2 \: + \: 8\,\alpha_3 \: + \: ... ... t} & = & (\alpha_4 \lambda) \; \mathrm e^{- \gamma_4 \lambda} .\end{eqnarray}$$ (A10) (A11)