Derivation of formulae for reduction in good events

Derivation of Event Loss Formula

If the conditions of a beam spill changes such that for the same beam intensity a greater fraction of bad events pass the trigger conditions, then a net loss in good events will be observed. This is because the spill length will be effectively shortened by the amount of time spent reading out the extra bad events. We shall derive an expression which relates the fractional loss of good events to the the measured "hadronic fractions" for good and bad spills.

Let us define:-

 Ng = No. of good events in the bad spill.
 Nb = No. of bad events expected if the "hadronic fraction" had not changed.
 Ne = No. of extra bad events due to the altered spill condtions.
 T  = Dead time due a single event.
 S  = Total spill length

The "hadronic fraction" for this spill is defined as:-

      HF = Ng/(Ng+Nb+Ne)                  (1)

For a good spill at the same beam intensity we would have expected a hadronic fraction given by:-

      HF'  = Ng/(Ng+Nb)                     (2)

We can also define the fractional dead time for the spill as follows:-

      DT  = T.(Ng+Nb+Ne)/S
 =>   T/S = DT/(Ng+Nb+Ne)                  (3)

Since, for fixed spill conditions, the number of good events obtained is proportional to the length of the spill, then the reduction in good events is given by:-

      R   = (S - T.Ne)/S
          = 1 - (T/S).Ne
          = 1 - DT.Ne/(Ng+Nb+Ne)           from (3)
          = 1 - DT.HF.Ne/Ng

Rearranging equation (1) we find :-

     1/HF = (Ng+Nb+Ne)/Ng
          = (Ng+Nb)/Ng + Ne/Ng
          = 1/HF' + Ne/Ng                  from (2)
 => Ne/Ng = 1/HF - 1/HF'
          = (HF'-HF)/(HF.HF')

Substituting this result into the expression for R gives:-


      R   = 1 - (HF'-HF).DT/HF'