1. CARIOCA Dead-time & inefficiency - UPDATE M2 station has been reanalysed from the Davide measurements at 20 MHz BC of counting rates on the bi-gap physical channel counters at 4 luminosities ~ (0.4 - 0.6 - 0.8 - 1) x 10 33 PROCEDURE 1)Measure inefficiency of particle counting 2)Extract the inefficiency fraction due to CARIOCA dead-time δ 3)Evaluate (by MC) the δ value generating such inefficiency 4)From this δ, evaluate the inefficiency we will have at 40 MHz All this per region/readout separately G. Martellotti, G. Penso, D. Pinci

2. DEFINITIONS: Ro = BC rate = 20MHz (in a 0.7 fraction of time - 1262 BC/orbit) Rpart = rate of particles hitting each Pad (not known) R*= rate of Pad counting (Pad not in dead-time) Note that the rates R D * measured per second by Davide have to be rescaled to R*=R D */0.7 when compared to Ro=20 MHz The inefficiency of the FE channel as particle counter is measured by the difference between the measured rate R* and the rate of hitting particles Rpart. Rpart is not known but it scales with luminosity L  2 measurements are made at L 1 and L 2 > L 1 The ratio ρ = (R 2 */ R 1 *) (L 1 / L 2 ) as a function of R 2 *- R 1 * measures with good approximation the inefficiency of a channel when the channel counting rate is R* = R 2 *- R 1 * Inefficiency of particle counting

3. The counting inefficiecy is due to 2 effects : # Non null probability of >1 particle hitting the same pad in the same BC (granularity effect)  rate of pad hit R < Rpart  this is not a detector inefficiency # When the pad is hit (rate R) it has a probability to be in the CARIOCA dead-time δ  R*=R(1-δR*)  this is a detector inefficiency If we want to evaluate the detector inefficiency (due to CARIOCA dead-time) from the counting rate, we have to ‘’subtract’’ the granularity effect calculating R starting from Rpart Counting efficiency Inefficiency of particle counting R* = R2* -R1* (GHz)

4. k = number of interactions in the BC  average number of interactions per BC Poissonian of k P k = exp(-  )  k / k! n = number of particles on the Pad  = average pad occupancy for 1 interaction when occurring k interactions in the BC = k  Poissonian of n P n = exp(-k  ) (k  ) n /n! Probability of pad hit in the case of k interactions = 1-exp(-k  ) Overall probability of Pad-on (1 – Po) =  k=1… exp(-  )  k / k! (1-exp(-k  )) = 1 - exp{-  [1-exp(-  )]}  A channel hit in average by  particles/BC will have a particle rate Rpart =  Ro and a lower rate of Pad-on R = (1–Po) Ro GRANULARITY EFFECT can be calculated in terms of  and 

5. Ro = 20 MHz, Rpart = μωRo  ω = Rpart/Roμ. We don’t know ω but it can be quite well approximated by ω ≈ R*/Roμ With this ω we calculate R = Ro (1-Po) and write the dead-time formula R*/R = (1 - δR*) Given two measurements at different luminosity we can calculate the ratio ρ = (R 2 */R 1 *) (R 1 / R 2 ) = (1 - δR 2 *)/(1 - δR 1 *) where R 1 and R 2 are calculated with the same ω and the two different μ 1, μ 2 Neglecting quadratic terms ρ ≈ 1 – δ(R 2 * - R 1 *)  Plotting ρ versus (R 2 * - R 1 *) we can directly measure δ Eventually, using this δ value, we can recalculate ω and make an iteration to improve the δ value

6. R 1 * measured at L 1 = 0.6 x 10 33 R 2 * measured at L 2 = 0.4 x 10 33 If we express the rate in GeV, the slope is expressed in nano seconds  effective dead-time δ eff due to the CARIOCA dead-time folded with the time BC sequence and the detector time response (bigap time response and dead-time variance) R* = R 2 * -R 1 * (GHz) inefficiency due to CARIOCA dead-time R* = R (1- δ eff R*) The inefficiency at R* is δ eff R* efficiency L1-L2

7. # Channels not having a meaningfull measurements for all the luminosities are eliminated # A few measurements too far from the mean value are eliminated # The average in bins of R 2 * – R 1 * can be performed  Linear Fit R* = R 2 * -R 1 * efficiency GHz SLOPE = effective dead-time δ eff = 48.5 ns Apparently a change in luminosity of few per mill

8. At a first look the slopes obtained for the other luminosity combinations appear to be consistent (a more detailed comparison is in progress) Luminosity = (4-6, 4-8, 4-10, 6-8, 6-10, 8-10) x 10 32 δ eff = (48.5, 50.1, 47.9, 48.6, 49.6, 46.9 )ns Average value δ eff ≈ 48±1.5 ns Structures are visible – to be interpreted as due to significantly different δ values for different cases (regions – wires – cathodes). We may also have slight changes of luminosity during the measurements of data samples used Next step is to analyse separately the different ‘’regions’’

9. Short lever arm when analysing only one region  Analyse the combination L1-L4 Nevertheless we can say that The slope δ eff is significantly different in different regions. It is higher for cathodes than for wires # This is due to the different capacitance but also to the hit quality (different average charge deposited) # slightly different extrapolations to zero. Possibly a change in luminosity of few per mill in between the measurements for wires and cathodes L1-L2 Region R1 δ eff 64ns δ eff 65ns δ eff 51ns δ eff 44ns efficiency

10. Even shorter lever arm for R2 Certainly too short for R3 R4 to measure a meaningful slope  Analyse L1-L4 δ eff R1 cathodes64-65 ns R1 wires44-51ns R2 cathodes53 ns R2 wires46-50ns L1-L2 Region R2 δ eff 53ns δ eff 50ns δ eff 53ns δ eff 46ns

11. Toy Monte Carlo Given # a value of CARIOCA dead time δcar (mean value and variance) # the time response of the bi-gap chamber σ t # the BC rate = 20 MHz or 40 MHz  Inefficiency due to δ is calculated (expressed in terms of effective dead time δ eff)

12. BCR = 20 MHz ASSUMED : bi-gap gaussian response σ t = 5ns δcar = 70 ± 11 ns (gaussian) Current BC Previous BCs contributing to dead time BC structure

13. BCR = 20 MHz σt = 4 ns σ δcar = δcar/10 σt = 0 σ δcar = 0 δeff (ns) δcar (ns) For each assumed (with its variance) MC  inefficiency Simple gaussian shapes have been assumed for the time fluctuations. The main contribution to the smoothing comes from the δcar variance assumed to increase with the δcar value. From the Riegler measurements I would say that the real behaviour is in between the two last hypoteses (nearer to the last one). inefficiency DT eletctronics (ns) σt = 5 ns σ δcar = δcar /6

14. BCR = 20 MHz δcar (ns) For each ‘’inefficiency’’ δeff measured in a region, δcar is evaluated If δeff ~ 50 ns the δcar is not very well determined. σ t = 5ns σ δcar = δcar/6 δcar (ns) δeff (ns) σ t = 4ns σ δcar = δcar/10 Average all regions R2 cathodes R1 cathodes

15. σ t = 5ns, σ δcar = δcar /6 For BCR = 40 MHz with any reasonable assumption on time fluctuations (in particular for the δcar variance) we will have a sufficient smoothing  inefficiency will be ~ proportional to CARIOCA dead-time - δeff ~ δcar -12.5 ns σ t = 4ns, σ δcar = δcar /10 δcar (ns) inefficiency δeff (ns)

16. Station M2 δeff (20 MHz)  δcarioca  δeff (40MHz) Average on all regions 48ns 67-73ns 54-61ns R1 cathodes 64-65ns 90-95ns 77-83ns R1 wires 44-51ns ? 60-80ns 47-68ns R2 cathodes 53ns 76-85ns 63-73ns R2 wires 46-50ns ? R3 ? R4 ? Inefficiency at a rate R (measured in GHz) is obtained from the dead-time formula R*= R(1 - δeff R*) A reliable error estimate is needed Further analysis is needed of the samples with larger lever arm: L1-L4 ‘’PROVISIONAL’’ Summary table

17. Summary # Inefficiency of physical channels due to CARIOCA dead time is measured from counting rates in terms of an effective δeff # preliminary results indicate that : 1) The average δeff is of the order of 50ns 2) δeff is significantly different for different regions/readouts. It is higher in R1 (and for cathodes is higher than for wires) – Assuming an average δcar seems to be a crude approximation (optimistic for the inner region) 3) From the δeff, the (larger) CARIOCA dead time (δcar) is evaluated with a toy MC making simple assumptions on its variance and on the bi-gap time response. 4) When going at 40 MHz BCR, we will have a (larger) inefficiency well approximated by a δeff ≈ δcar – 12.5 ns  The inefficiency of R1 cathodes is very high. At the high rates foreseen for future, it seems that inefficiency will be too high to work with the END of wires and cathodes # With the muon FE counters, carefull measurements would allow to monitor relative luminosities with very high precision