6/9/2016W. Riegler1 Analytic Expressions for Time Response Functions (and Electric Fields) in Resistive Plate Chambers Werner Riegler, CERN k=0 k=0.6

6/9/2016W. Riegler2 Analytic Models of the Time Response Function A. Mangiarotti, A. Gobbi, On the pysical origin of tails in the time response of spark counters, NIM A 482 (2002) W. Riegler, C. Lippmann, R. Veenhof, Detector physics and simulation of resistive plate chambers, NIM A 500 (2003) A. Gobbi, A. Mangiarotti, The time response function of spark counters and RPCs, NIM A 508 (2003) B. Blanco et al., Resistive plate chambers for time of flight measurements, NIM A 513 (2003) 8-12 C. Lippmann, W. Riegler, Space charge effects in resistive plate chambers, NIM A 517 (2004)  Spacecharge effects already present at the threshold level but the time resolution is unaffected. A. Mangiarotti, P. Fonte, A.Gobbi, Exacly solvable model for the time response function of RPCs, NIM A 533 (2004) W. Legler, Die Statistik der Elektronenlawinen in elektronegativen Gasen, bei hohen Feldstärken und bei grosser Gasverstärkung, Z. Naturforschg. 16a, (1961)  Avalanche multiplication in electronegative gases. The processes of primary ionization, avalanche multiplication and signal induction in RPCs are by now well understood and can be formulated with simple Monte Carlo programs, and it is straight forward to simulate the RPC time response function. An analytic expression has the advantage of giving a deeper insight into the dependence on parameters and the limits of achievable performance. Analytic models of the time response function with ever increasing detail were developed over the last years.

6/9/2016W. Riegler3 E0E0 Electron avalanches doesn’t cross the threshold Efficient Gap: Electron avalanches cross the threshold Efficient clusters Primary Ionization and Avalanche Multiplication Number of clusters per unit length follows strictly a Poisson distribution. Number of efficient clusters follows to a good approximation the same Poisson distribution. The number of electrons per cluster follows approximately a 1/n 2 distribution.  Number of efficient electrons follows approximately a Landau distribution. Each individual electron starts and avalanche, inducing a signal which will cross a given threshold of the readout electronics  time.

6/9/2016W. Riegler4 Analytic Models of the Time Response Function The most detailed model so far (A. Mangiarotti, P. Fonte, A. Gobbi ) ρ(n,t) = probability that the number of electrons in the gas gap crosses a threshold of n electrons between t and t+dt S =(α-η)v k = η/α n 0 = Average number of efficient clusters fluctuation according to a Poisson distribution One electron per cluster Large n approximation of Legler’s avalanche model I 1 (x) … modified Bessel function of first kind Remarkable: The explicit inclusion of the electron attachment doesn’t change the shape of the time response function but is just changing the number of efficient clusters. Example: k=0.2, n0=3, n=100

6/9/2016W. Riegler5 n 0 = Average number of efficient clusters n av = Average number of electrons per cluster The r.m.s. time resolution is therefore a universal function of n eff E.g. (α-η)=100[1/mm], v=0.2[mm/ns], n eff = 3 S=20[1/ns], r.m.s=1/20[ns] = 50[ps] Analytic Models of the Time Response Function It turns out that this function is even correct for a cluster size distribution following a power law ! p(n) = probability o find n efficient clusters, f(n) = probability to find n electrons per cluster π 2 /6 α 1/√n eff A. Mangiarotti, P. Fonte, A. Gobbi r.m.s * S

6/9/2016W. Riegler6 Cluster Size Distribution The number of electrons per cluster is however not distributed according to a power law. The simplest model respecting elementary physics is a 1/n 2 cluster size distribution, based on the 1/E 2 crossection that the incoming particle transfers the energy E to and electron. 1/n 2 Power Law n av =2.7 Simulation with HEED for C 2 F 4 H 2 /i-C 4 H 10 /SF 6 85/5/10

6/9/2016W. Riegler7 Primary Electron Distribution The Poissonian cluster number distribution and 1/E 2 energy transfer probability result in the Landau distribution P(E) for the energy loss in the gas gap. The Poissonian cluster number distribution and 1/n 2 cluster size distribution result a discrete analog to the Landau distribution. If n 0 denotes the average number of efficient clusters, the probability g(m) to find m efficient electrons in the gas gap is

The probability g(m) to find m efficient electrons in the gas gap, assuming a Poisson distribution of efficient clusters with an average of n 0 and a power law cluster size distribution with average of n av is where 1 F 1 (a,b;z) is the Kummer confluent hypergeometric function. The probability g(m) to find m efficient electrons in the gas gap, assuming a Poisson distribution of efficient clusters with an average of n 0 and a 1/n 2 cluster size distribution is 6/9/2016W. Riegler8 Primary Electron Distribution Simulation with HEED for C 2 F 4 H 2 /i-C 4 H 10 /SF /3/0.3 Power Law n av =2.7 1/n efficient electrons

If g(m) is the probability that an incoming particle leaves m efficient electrons in the gas gap and h(m,n,t)dt is the probability that an avalanche starting with m electrons crosses a threshold of n electrons between time t and t+dt, the time response function ρ(n,t) is given by If G(z) is the Z-transform of g(m) and H(z,n,t) is the Z-transform of h(m,n,t), the time response function can be expressed as where z is a complex variable and the path of integration is a circle with radius r centered at zero. Parametrizing the circle of integration we can write this in a form that is well suited for numerical evaluation: 6/9/2016W. Riegler9 General Analytic Expression for the Time Response Function

6/9/2016W. Riegler10 Primary electron Distribution, g(m), G(z) For a fixed number of m efficient electrons we have If the number of efficient cluster is Poisson distributed with an average of n 0 and the probability to have n electrons per cluster is f(n), with Z transform F(z), the probability to have n efficient electrons is given by For a single electron per cluster we have For a power law cluster size distribution we have For a 1/n 2 cluster size distribution we have

Legler’s avalanche model assumes a probability αdx that an electrons duplicates over a distance dx and a probability ηdx that an electron attaches within a distance dx, independent of the electron’s history. This leads to the partial differential-difference equation The exact solution for H(z,n,t) is: For n>>1 the expression can be approximated by In the following we use this continuous approximation which is definitely adequate for avalanche sizes in RPCs. The exact expression for H(z,n,t) will be used later to see at what electron numbers the time response function becomes independent of the applied threshold. 6/9/2016W. Riegler11 Avalanche Multiplication h(m,n,t), H(z,n,t)

The normalized time response function for an arbitrary distribution of efficient electrons g(m), G(z) and assuming the large n approximation of Legler’s avalanche model is therefore Before explicitly calculating ρ(n,t) for the previously mentioned electron distributions we investigate three remarkable properties of ρ(n,t) which hold for any G(z): 1)Threshold independence of the time resolution 2)Cluster transformation 3)Attachment transformation 6/9/2016W. Riegler12 Time Response Function

6/9/2016W. Riegler13 Time Response Function Properties: 1) Threshold independence If we multiply the threshold n of the time response function by a factor β = exp (ln β), it transforms into Scaling the threshold by β results therefore only in a shift of the time response function by T= - ln β/S without changing it’s shape.  The RPC time resolution is independent of the applied threshold.

6/9/2016W. Riegler14 Time Response Function Properties: 2) Cluster Transformation If we transform z, the variable of integration of the time response function according to with 0<a<1, the time response function transforms into The time response function for a electron distribution G(z) and attachment k is therefore equal to the time response function for a electron number distribution G’(y)=G((y-a)/(1-a)) and attachment k’. For example, the time resolution for a single efficient electron with attachment k is equal to the time resolution for an efficient electron distribution following a power law with k’

6/9/2016W. Riegler15 In the absence of attachment, the time response function takes the form (k=0) By transforming the above integral according to the time response function transforms into The time response function for an electron distribution G(z) and attachment k is therefore equal to the time response function for an electron distribution G(y/(1-k+ky)) in absence of attachment. Time Response Function Properties: 3) Attachment Transformation

The time response function has an average value given by The dependence on the threshold n is simply a simply shift by T=ln n/S. The variance is given by which is independent of the threshold n. 6/9/2016W. Riegler16 Average and Variance

6/9/2016W. Riegler17 Example 1: Fixed Number of m Efficient Electrons k=0 k=0.3 k=0.6k=0.9 The integral can be evaluated by finding the residual at z=0 which results in

6/9/2016W. Riegler18 Example 1: Fixed Number of m Efficient Electrons k=0 k=0.3 k=0.6 k=0.9

Curiosity: In absence of attachment the variance of ρ(n,t) is We know that for increasing the number m of primary electrons, the variance, i.e. the RPC time resolution, becomes zero, which means that we have We have therefore answered the famous ‘Basel problem’ of finding the sum of all inverse squares, which was solved by Euler in 1735, by ‘measuring’ the RPC time resolution ! To be honest – for this argument we used the theory of complex functions and he central limit theorem which were only discovered in the late 19 th century … 6/9/2016W. Riegler19 Example 1: Fixed Number of m Efficient Electrons

6/9/2016W. Riegler20 Example 2: Single Electron or Power Law Cluster Size Assuming k=0 and finding the residual at z=0, the evaluation of the integral is straight forward. Using the cluster transformation (2) and attachment transformation (3) properties, the time response function for attachment k and a power law cluster size distribution becomes Start with k=0 and a single electron per cluster:

6/9/2016W. Riegler21 Example 3: 1/n 2 Cluster Size Distribution Integration must be performed numerically Power law cluster size distribution 1/n 2 cluster size distribution The higher probability of large clusters for the 1/n 2 cluster size distribution increases the tail towards earlier times and makes the time response function more symmetric. The r.m.s. is larger.

6/9/2016 W. Riegler 22 Example 4: HEED Cluster Size Distribution Integration must be performed numerically The higher probability of large clusters for the 1/n 2 cluster size distribution increases the tail towards earlier times and makes the time response function more symmetric. The r.m.s. is larger. Including the detailed cluster size distribution calculated with HEED reduces the effect. Power law cluster size distribution 1/n 2 cluster size distribution HEED

6/9/2016W. Riegler23 1/n 2 Power Law HEED Comparison

6/9/2016W. Riegler24 R.M.S time resolution Time resolution for power law and 1/n 2 model for k=0, 0.3, 0.6 Power law 1/n 2 k=0.6 k=0.3 k=0 Ratio of 1/n 2 and power law model for k=0, 0.3, 0.6  Expected r.m.s is % larger. k=0 k=0.3 k=0.6

6/9/2016W. Riegler25 R.M.S time resolution Time resolution for power law and 1/n 2 model for k=0, 0.3, 0.6 Power law 1/n 2 k=0.6 k=0.3 k=0 Ratio of HEED and power law model for k=0, 0.3, 0.6  Expected r.m.s is 20-50% larger. k=0 k=0.3 k=0.6

6/9/2016 W. Riegler26 n=5n=15 n=50n=100 n=300 Exact Solution of Avalanche model for n 0 =4 and Different Thresholds n For thresholds corresponding to n>1000 electrons the continuous approximation of Legler’s avalanche model is perfectly adequate and the time resolution is independent of the threshold.

We have developed a formalism that allows the calculation of electric fields in any RPC geometry, i.e. the (time dependent) electric fields of point charges in a geometry of N parallel layers of a given dielectric permittivity and conductivity. The solution is given by a potential for of each of the N layers Knowing the potential we can calculate 1)Space charge effects 2)Induced signals on rectangular pads or strips 3)Electric Field fluctuations due to randomness of particle flux 4)HV electrode transparencies 5) Signal crosstalk 6/9/2016W. Riegler27 Electric Fields in Resistive Plate Chambers

General solution in Cylindrical coordinates General solution in Cartesian coordinates with where A n and B n are the solution of the following 2Nx2N matrix equation 6/9/2016W. Riegler28 Electric Fields in Resistive Plate Chambers, General Solution

6/9/2016W. Riegler29 Electric Fields Fluctuations M. Abbrescia, MC simulation of field fluctuations assuming and effective area A for a cell capacitor model. RPC2005: 1) D. Gonzalez, Analytic Expression of field fluctuation assuming and effective area A for a cell capacitor model. 2) C. Lippmann, W. Riegler, MC simulation of field fluctuation assuming the exact solution for the electric field Unification: Analytic expression for effective area from exact solutions  1)+2) = analytic expression for field flucutuations !

6/9/2016W. Riegler30 T<<  T=  T=10  T=50  T=500  I 1 (t) I 3 (t) I 5 (t)  =  0 /  The conductive layer ‘spreads’ the signals across the strips. Electrode Transparency and Crosstalk

6/9/2016W. Riegler31 An analytic expression for the RPC time response function, allowing the inclusion of realistic electron cluster size distributions in order to respect elementary physics processes, was presented. Compared to models assuming a single electron per cluster or a power law cluster size distribution, the time response function is more ‘symmetric’ and the expected r.m.s. time resolution is 20-50% worse. The expression is well suited for numerical evaluation of detailed primary electron distributions, which avoids running lengthy MCs. A general solution for electric fields in RPCs was developed, which allows the calculation of space charge effects, weighting fields, induced signals, field fluctuations and crosstalk. With all this material at hand we are finally in a position to include RPC simulation into the detector simulation program GARFIELD, which is planned during Conclusion