Estimation of accidentals for pi0 experiment. Accidentals here are defined as events in which un-related background processes produce a signal similar to those accepted by the trigger. Specifically, this would be events in which two particles hit the calorimeter (each having enough energy to fire the discriminators) close enough in time to create a coincidence in the electronics. For the purposes of this document, gate widths will be assumed as "effective" gate widths which would be slightly smaller that the actual gate widths set so as to account for the neccessary overlap time in the coincidence unit. Let the discriminator pulse width from a group of calorimeter elements (20-24 elements per group) be denoted w1 Let the rate for single background beam events which fire one of the discriminators in the calorimeters be R1. It is useful to break the singles rate up into two equal parts when considering random coincidences. The sum is then still just R1, but the two rates can be thought of as separate. If an event from the first rate fires, an event from the second rate will fire (roughly speaking) once in the 1/(R1/2) seconds surrounding the event from the first rate. There are only (w1+w1) seconds inside of that window in which a coincidence can occur. Thus, the probability of the second rate having an event in coincidence with the first (given the first occured) is given by: (w1+w1) ---------- = w1 * R1 1/(R1/2) Multiplying by the rate of the first gives the rate of accidental coincidences. R2 = w1 * R1 * (R1/2) The triggering scheme, however will look for coincidences between two quarters of the detector and will not fire when only one quarter is hit. Only three quarters of the events in R2 therefore would produce valid triggers 3 * w1 * R1 * R1 Acc = R2 * 3/4 = ----------------- 8 where: w1 is the pulse width of a single calorimeter discr. R1 is the total singles rate in the calorimeter The trigger will also require a coincidence with the tagger master OR (MOR). In principle, a single background event should in coincidence with the tagger. For accidental coincidences between two beam-related background events, the probability of the tagger firing for at least one of them is approximately 1. =============================================================== SIMULATION =============================================================== A GEANT simulation was done to estimate a value for the singles rate (R1) as a function of threshold on the group discriminators. The simulation generated 101M bremmstrahlung photons between 100MeV and 6GeV (the nominal electron beam energy). The simulation thus represents 8 3.61 x 10 7 ------------------ = 8.82 x 10 equivalent photons ln(6GeV/100MeV) 7 With a nominal beam intensity of 7.2 x 10 eq. ph./sec the simulation represents 7 8.82 x 10 ----------- = 1.22 seconds of beam time 7 7.2 x 10 The number of events as a function of minimum energy deposited in the detector can be seen in the following table: Ndep=1046115 Nphotons=360519488 beam_time=1.22296 low_thresh_rate=855396 thresh=0.199 n=128566 rate=41.4441 thresh=0.399 n=42286 rate=4.48331 thresh=0.599 n=17630 rate=0.779309 thresh=0.799 n=8179 rate=0.167729 thresh=0.999 n=3997 rate=0.0400566 thresh=1.199 n=2165 rate=0.0117523 thresh=1.399 n=1227 rate=0.00377481 thresh=1.599 n=739 rate=0.00136929 thresh=1.799 n=483 rate=0.000584925 thresh=1.999 n=325 rate=0.000264833 thresh=2.199 n=224 rate=0.000125807 thresh=2.399 n=161 rate=6.4992E-05 thresh=2.599 n=117 rate=3.43222E-05 thresh=2.799 n=92 rate=2.12218E-05 thresh=2.999 n=66 rate=1.09218E-05 also given in the above table are the rates for accidental coincidences. The rates are caluculated from the above equation for Acc using a pulse width of 10 ns and using n/1.22 secs for R1. A second source of background beam realated events is from processes which produce two energetic particles in the calorimeter which are both associated with a single bremmstrahlung photon. The simulation produced 6 events in which two quadrants of the detector had >1GeV deposited in them. Since the 1 GeV may not have come from a single particle, the engery may have been spread over the entire quadrant. This calculation should therefore be considered an upper limit. 6 ------ = 5 +/- 2 Hz Upper limit 1.22 The above rate calculation does not include consideration of the tagging energy range. Looking specifically for events in which two particles (from the same photon) both greater than 1 GeV hit the target in different quadrants, only one event was found with an initial photon energy in the tagging range. This leads to a lower limit of 1 ------ = 1 +/- 1 Hz Lower limit 1.22