ιατµηµατικό Πρόγραµµα Μεταπτυχιακών Σπουδών Φυσική και Τεχνολογικές Εφαρµογές Μελέτη των resistive layers και προσοµοίωση ανιχνευτων Micromegas

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1 ΕΘΝΙΚΟ ΜΕΤΣΟΒΙΟ ΠΟΛΥΤΕΧΝΕΙΟ ΣΧΟΛΗ ΕΦΑΡΜΟΣΜΕΝΩΝ ΜΑΘΗΜΑΤΙΚΩΝ & ΦΥΣΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΣΧΟΛΗ ΜΑΧΑΝΟΛΟΓΩΝ ΜΗΧΑΝΙΚΩΝ ΕΚΕΦΕ ΗΜΟΚΡΙΤΟΣ ΙΝΣΤΙΤΟΥΤΟ ΝΑΝΟΕΠΙΣΤΗΜΗΣ & ΝΑΝΟΤΕΧΝΟΛΟΓΙΑΣ ΙΝΣΤΙΤΟΥΤΟ ΠΥΡΗΝΙΚΗΣ & ΣΩΜΑΤΙ ΙΑΚΗΣ ΦΥΣΙΚΗΣ ιατµηµατικό Πρόγραµµα Μεταπτυχιακών Σπουδών Φυσική και Τεχνολογικές Εφαρµογές Μελέτη των resistive layers και προσοµοίωση ανιχνευτων Micromegas ΜΕΤΑΠΤΥΧΙΑΚΗ ΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ του Ιωάννη ρίβα Κουλούρη Επιβλέπων : Θεόδωρος Αλεξόπουλος Καθηγητής Ε.Μ.Π. ΑΘΗΝΑ Ιούνιος 19

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3 iii ΕΘΝΙΚΟ ΜΕΤΣΟΒΙΟ ΠΟΛΥΤΕΧΝΕΙΟ ΣΧΟΛΗ ΕΦΑΡΜΟΣΜΕΝΩΝ ΜΑΘΗΜΑΤΙΚΩΝ & ΦΥΣΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΤΟΜΕΑΣ ΦΥΣΙΚΗΣ ΕΡΓΑΣΤΗΡΙΟ ΠΕΙΡΑΜΑΤΙΚΗΣ ΦΥΣΙΚΗΣ ΥΨΗΛΩΝ ΕΝΕΡΓΕΙΩΝ & ΣΥΝΑΦΟΥΣ ΟΡΓΑΝΟΛΟΓΙΑΣ Μελέτη των resistive layers και προσοµοίωση ανιχνευτων Micromegas ΜΕΤΑΠΤΥΧΙΑΚΗ ΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ του Ιωάννη ρίβα Κουλούρη Επιβλέπων : Θεόδωρος Αλεξόπουλος Καθηγητής Ε.Μ.Π. Εγκρίθηκε από την τριµελή εξεταστική επιτροπή στις.... Θ. Αλεξόπουλος Καθηγητής Ε.Μ.Π.... Ε. Γαζής Καθηγητής Ε.Μ.Π.... Σ. Μαλτέζος Αν. Καθηγητής Ε.Μ.Π. Αθήνα, Ιούνιος 19

4 iv... Ιωάννης ρίβας Κουλούρης Φυσικός Εφαρµογών c (19) Εθνικό Μετσόβιο Πολυτεχνείο. All rights reserved. Απαγορεύεται η αντιγραφή, αποθήκευση και διανοµή της παρούσας εργασίας, εξ ολοκλήρου ή τµήµατος αυτής για εµπορικό σκοπό. Επιτρέπεται η ανατύπωση, αποθήκευση και διανοµή για σκοπό µη κερδοσκοπικό, εκπαιδευτικό ή ερευνητικής ϕύσεως, υπό την προϋπόθεση να αναφέρεται η πηγή προέλευσης και να διατηρείται η παρούσα σηµείωση. Ζητήµατα που αφορούν την εκτίµηση της εργασίας για κερδοσκοπικό σκοπό πρέπει να απευθύνονται προς τον συγγραφέα. Οι απόψεις και τα συµπεράσµατα που περιέχονται σε αυτή τη δήλωση εκφράζουν τον συγγραφέα και δεν πρέπει να ϑεωρηθεί ότι αντιπροσωπεύουν τις επίσηµες ϑέσεις του Εθνικού Μετσόβιου Πολυτεχνείου.

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6 Περίληψη Η παρούσα διπλωµατική εργασία εκπονήθηκε στα πλαίσια της ολοκλήρωσης των µεταπτυχιακών µου σπουδών και αποτελεί µια εισαγωγή της µελέτης των ανιχνευτών Micromegas. Στο πρώτο κεφάλαιο γίνεται µια αναφορά στους ανιχνευτές αερίου και κυρίως στους ανιχνευτές MicroMegas.Ο ανιχνευτής MicroMegas ανήκει στην κατηγορία των ανιχνευτών αερίου, έχει µικρό κόστος, παρουσιάζει αντοχή σε περιβάλλον υψηλής ακτινοβολίας και παρέχει πολύ καλή χωρική και ενεργειακή διακριτική ικανότητα. Εχει χρησι- µοποιηθεί σε πολλά πειραµατα στο CERN έχει επιλεχθεί για την αναβάθµιση του New Small Wheel ϑαλάµου των µιονίων του πειράµατος ATLAS. Στο δεύτερο κεφάλαιο εργαζόµαστε πάνω στα στατικά και χρονοεξαρτηµένα ηλεκτρικά πεδία κάποιων ανιχνευτών µε παράλληλα στρώµατα µε δοσµένη ορατότητα και αδύναµη αγωγιµότητα. ουλεύουµε πάνω στο πεδίο ενός σηµειακού ϕορτίου, καθώς και στα weighting fields για readout pads και readout strips αυτές τις γεωµετρίες. Μελετάµε πως η διάδοση ενός ϕορτίου επηρεάζει τα στρώµατα µε αντίσταση. Επίσης προσπαθούµε να διερευνήσουµε την επίδραση της ογκώδους αντίστασης στα ηλεκτρικά πεδία και τα σήµατα. Εφαρµόζουµε τα αποτελέσµατα για να εξάγουµε πεδία και επαγόµενα σήµατα σε Resistive Plate Chambers, ανιχνευτές MicroMegas που περιλαµβάνουν στρώµατα αντίστασης για διασπορά ϕορτίου και προστασία από την εκφόρτιση. Αναλύουµε επίσης λεπτοµερώς πως επηρεάζουν τα Resistive layers τα σχήµατα των σηµάτων και αυξάνουν το crosstalk µεταξύ των readout electrodes. Στο τρίτο κεφάλαιο πραγµατοποιείται η προσοµοίωση διαφόρων µοντέλων ανιχνευτών MicroMegas ANSYS Maxwell µε στόχο την παρατήρηση και τον υπολογισµό της χωρητικότητας µεταξύ των strips. τη χρήση του Στο τέταρτο κεφάλαιο πραγµατοποιείται η προσοµοίωση διάφορων µοντέλων ανιχνετών MicroMegas µέσω του LTspice και παρατηρήθηκε το σήµα εξόδου του κεντρικόυ strip και των γειτονικών του. Επίσης παρατηρήθηκε πως αλλάζει το σήµα µε το να µεταβάλλουµε την χωρητικότητα µεταξύ των resistive strips, των readout strips και τη χωρητικότητα µεταξύ readout-resistive strips. vi

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8 Abstract The present diploma thesis is an introduction of the study of MicroMegas. In section 1 we give a report to Gaseous detectors and specially to MicroMegas detectors. MicroMegas is a gaseous detector, with a low construction cost, a tolerance at high radiation environment and a very good spatial and energy resolution. This technology has been used so far in many experiments at CERN and is chosen for the New Small Wheel upgrade at the ATLAS experiment. In section we work on the static and time dependent electric fields in some detectors geometries with parallel layers of a given perimittivity and weak conductivity. We work on the field of a point charge,as well as the weighting fields for Readout pads and Readout strips in these geometries.we investigate how the spreading of the charge effect the Resistive layers.we also try to investigate the effect of "bulk" Resistivity on electric fields and signals.we apply the results to derive fields and induced signals in Resistive Plate Chambers,MicroMegas detectors including Resistive layers for charge spreading and discharge protection. We also discuss in details how the Resistive layers affect signal shapes and increase the crosstalk between readout electrodes. In section 3 takes place the simulation of a variety of Modules of MicroMegas detectors with the help of AN- SYS Maxwell,to observe and calculate the capacitance between the strips. In section 4 takes place the simulation of a variety of Modules of Micromegas detector with LTspice. With spice were taken the figures of the output signal for the central and the neighbor strips. It also observed how this signal changes if we change the capacitance between the Resistive strips, the Readout strips and the capacitance between Readout-Resistive strips. viii

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10 Ευχαριστίες Με την ολοκλήρωση της παρούσας διπλωµατικής εργασίας ϑα ήθελα να εκφράσω τις ειλικρινείς µου ευχαριστίες σε όλους όσους µε ϐοήθησαν και στήριξαν σε όλη την διάρκεια της. Αρχικα ϑα ήθελα να ευχαριστήσω την επιβλέποντα καθηγητή µου, κ. που µου έδωσε, καθώς και για την καθοδήγηση και τη ϐοήθεια του. Θεόδωρο Αλεξόπουλο για την εµπιστοσύνη Θα ήθελα να ευχαριστήσω τον Αν. Καθηγητή Σταύρο Μαλτέζο για τις γνώσεις που προσφέρει πάντα µε χαρά στους ϕοιτητές πάνω στη ϕυσική και στα ηλεκτρονικά. Θα ήθελα να ευχαριστήσω τη διδάκτορα Χαρά Κιτσάκη για τη πολύτιµη ϐοήθεια που µου προσέφερε στην αρχή της εργασίας, αλλα και τις συµβουλές της µέχρι την ολοκλήρωση της, καθώς και για τις συζητήσεις που είχαµε κατά τη διάρκεια των ωρών που ϐρισκόµασταν στο γραφείο. Θα ήθελα να ευχαριστήσω επίσης το διδάκτορα Αιµίλιο Κουλούρη για τη ϐοήθεια του πανω στα προγράγραµµα προσοµοίωσης, αλλα και τον µεταπτυχιακό ϕοιτητή Πολυνείκη Τζανή για τη ϐοήθεια του πάνω σε τεχνικά ϑέµατα της εργασίας. Τέλος ϑα ήθελα να ευχαριστήσω τους γονείς µου Κωνσταντίνο και Κωνσταντίνα για την υποστήριξη και αγάπη τους. x

11 Contents List of Figures xiv 1 Gaseous Detectors Avalanche Gaseous Detectors Micromegas Electric fields,weighting fields and signals in detectors including resistive materials Potential of a point charge centered at the origin Dicharges on a resistive MICROMEGA Potential of a point charge in a geometry grounded on a rectangle Weighting Fields N-Layer geometry Single Layer RPC Single Thin Resistive Layer Uniform currents on thin resistive layers Uniform Currents on a resistive plate covered with a thin resistive layer Signals and charge spread in detectors with resistive elements Layer with surface resistivity Micromegas Simulation with Ansys Maxwell Structure of the creating Modules by Maxwell Micromegas Modules More cases of micromegas Modules Coclusion Micromegas Simulation with LTspice Introduction Spice Simulation with LM and SM Modules Spice with Mesh Changes on Capacitances Appendices 63 A Changes to Capacitances between Resistive Strips 64 B Changes to Capacitances between Readout Strips 71 C Changes to Capacitances between Resistive-Readout strips 77 xi

12 List of Figures 1 Avalanche in the shape of drop The different regions of gaseous detectors operation with respect to the applied voltage Micromegas Two areas of electric field Total auxiliary current as a function of time. The peak at the origin lasts for less than ns is due primarily to the electrons movement while after ns the distribution id due to the ion drifting towards the mesh Sketch of the detector principle(not the scale), illustrating the resistive protection theme A point charge Q on the boundary between two dielectric layers Charge Q for different values of voltage Values of charge for different values of distance between mesh and the resistive strips for applied voltage 57 V A point charge Q in an empty condenser rectangular readout pad A geometry on N dielectric layers enclosed by grounded metallic plates. On the boundary between two layers at r = there are point charges Q A geometry with 3 dielectric layers Geometry with three layers and one point charge representing a single gap RPC Weighting field E z at position z = g/ for b = 4g and w x = g. The three curves represent ɛ r = 1(bottom),ɛ r = 8(middle),ɛ r = (top) Weighting field for a strip electrode of width w x and infinity extension A geometry with three layers and one point charge Current density i (r) at z = b. The blue curve represent the second order approximation of (.74), the green curve the fourth order approximation of (.74) and the yellow curve the approximation of (.75) A point charge placed on a resistive layer that is grounded on a rectangle A point charge placed on a resistive layer that is grounded on at x = and x = a but insulated on the others border Currents I 1x (t)(top) and I x (t)(bottom) from the of figure 3 for x = a/4. The straight line in the middle refers to the approximation from Uniform current "impressed" on the resistive layer will result in a potential distribution that depends strongly on the boundary conditions Voltage across the center of the resistive plate for a value of f = d/(a) =.1. The dots refer to the exact formula (.15), the curved line corresponds to the approximation from (.16) Uniform charge movement from for z = to z = g, with ɛ r = 1,w x = 4g,b = g,t = 1T Uniform charge movement from for z = to z = g, with ɛ r = 1,w x = 4g,b = g,t = T Uniform charge movement from for z = to z = g, with ɛ r = 1,w x = 4g,b = g,t =.1T Uniform charge movement from for z = to z = g, with ɛ r = 1,w x = 4g,b = g,t = 1T Uniform charge movement from for z = to z = g, with ɛ r = 1,w x = 4g,b = g,t = T Uniform charge movement from for z = to z = g, with ɛ r = 1,w x = 4g,b = g,t =.1T Uniform charge movement from for z = to z = g, with ɛ r = 1,w x = 4g,b = g,t =.1T Uniform charge movement from for z = to z = g, with ɛ r = 1,w x = 4g,b = g,t =.1T Micromegas Module by microscope Micromegas Module with 3 Resistive and 3 readout strips xii

13 xiii List of Figures 39 LM Module SM Module strips LM Module Table of capacitance for 5 strips LM Module strips SM Module Table of capacitance for 5 strips SM Module strips LM Module Table of capacitance for 9 strips LM Module strips SM Module Table of capacitance for 9 strips SM Module strips 1 µm LM Module Table of capacitance for 5 strips 1 µm LM Module strips LM Module with glue Table of capacitance for 5 strips SM Module with glue strips SM Module with glue Table of capacitance for 5 strips SM Module with glue Ludwig-Maximilians-Universitat Munchen - Lehrstuhl Schaile Module strips LM Module with Spice output voltage for central and neighbor strips output current for central and neighbor strips strips SM Module with Spice output voltage for central and neighbor strips output current for central and neighbor strips strips LM Module with Spice output voltage for central and neighbor strips output current for central and neighbor strips strips SM Module with Spice output voltage for central and neighbor strips output current for central and neighbor strips Circuit with a Mesh and a resistance Value of percentage due to capacitance between Resistive strips Value of percentage due to capacitance between Resistive strips Value of percentage due to capacitance between readout strips Value of percentage due to capacitance between readout strips Value of percentage due to capacitance between Resistive-Readout strips Value of percentage due to capacitance between Resistive-Readout strips A.1 1 pf between Resistive strips A. 3 pf between Resistive strips A.3 5 pf between Resistive strips A.4 1 pf between Resistive strips A.5 15 pf between Resistive strips A.6 pf between Resistive strips A.7 5 pf between Resistive strips A.8 3 pf between Resistive strips A.9 35 pf between Resistive strips A.1 1 pf between Resistive strips A.11 3 pf between Resistive strips A.1 5 pf between Resistive strips A.13 1 pf between Resistive strips A pf between Resistive strips A.15 pf between Resistive strips A.16 5 pf between Resistive strips A.17 3 pf between Resistive strips A pf between Resistive strips B.1 3 pf between Readout strips B. 5 pf between Readout strips B.3 1 pf between Readout strips B.4 15 pf between Readout strips

14 List of Figures xiv B.5 pf between Readout strips B.6 5 pf between Readout strips B.7 3 pf between Readout strips B.8 35 pf between Readout strips B.9 3 pf between Readout strips B.1 5 pf between Readout strips B.11 1 pf between Readout strips B.1 15 pf between Readout strips B.13 pf between Readout strips B.14 5 pf between Readout strips B.15 3 pf between Readout strips B pf between Readout strips C.1 1 pf between Resistive and Readout strips C. 3 pf between Resistive and Readout strips C.3 5 pf between Resistive and Readout strips C.4 1 pf between Resistive and Readout strips C.5 15 pf between Resistive and Readout strips C.6 pf between Resistive and Readout strips C.7 5 pf between Resistive and Readout strips C.8 3 pf between Resistive and Readout strips C.9 35 pf between Resistive and Readout strips C.1 1 pf between Resistive and Readout strips C.11 3 pf between Resistive and Readout strips C.1 5 pf between Resistive and Readout strips C.13 1 pf between Resistive and Readout strips C pf between Resistive and Readout strips C.15 pf between Resistive and Readout strips C.16 5 pf between Resistive and Readout strips C.17 3 pf between Resistive and Readout strips C pf between Resistive and Readout strips

15 1 1. Gaseous Detectors 1 Gaseous Detectors 1.1 Avalanche Gas Multiplication One of the most important phenomenon that happens on the gaseous detectors is the Avalanche. The primary electrons, so called the first ionisation products, drift to the electrodes of the anode through the gas. When their energy exceeds a sufficient value due to the affection of the field, they can create further ionizations in the gas. Secondary electrons, if the energy is sufficient, will do the same and this process goes on. So we have a production of ion-electron pairs and the formation of the avalanche. The electrons drifts faster and so the avalanche has the shape of a drop (figure 1). In the front we have the side electrons are moving along and on the back we have the slower electrons. At the end there is a large number of electrons in the cathode, that can easily be detected by electronic devices Figure 1: Avalanche in the shape of drop Avalanche depends on electric field that is applied on gas chamber and the pressure of the gas, which affects the free path of the electrons. If we consider λ as the mean free path of the electrons between two collisions then the coefficient a = 1/λ is the ionization probability per length. For a number of n electrons a path dx we will have further dn electrons : dn = nadx (1.1) So if we derive the number n of electrons, in the path x is : n = n e ax (1.) with a the ionization probability : a = P Ae BP E (1.3) with E the intensity of the electric field and A,B two constants depending on the gas in unit of cm 1 T orr 1 and V cm 1 T orr 1 respectively. Back to the number of electrons we can define M such the multiplier factor : M = n n = e ax (1.4) The M factor is help to find a good approximation for the number of electrons that reach the anode, where we read the signal. The multiplier factor M has a limit of M = 1 8.

16 1. Gaseous Detectors In figure are represent for a variety of gases the ionization probability depending on energy and the dependence of the factor a/p due to E/P. (a) Ionization Probability (b) Townsend Coefficient Figure 1.1. The role of photons During the multiplication process some electrons, that gained enough energy, instead of ionizing further the gas molecules, may bring them in an excited state. These excited molecules do not contribute directly to the avalanche but decay their ground state through the emission of a visible ultraviolet photon. Under some those photons can create ionizations in the gas with the limit of M = 1 8. These poly-atomic gases called the quench gases Signal Formation During the avalanche formation, a number of electron-ions pairs is created. The electrons and the positive ions, separated by the electric field, drift towards different directions. Their motion within the gas volume induce charge on electrons. Electrons drift fast and within a few nanoseconds reach the anode. Therefore, the current flowing on the electrodes is on the order of the nanoseconds and usually ignored by the detector electronics. On the other hand the positive ions drift with a velocity two to three orders of magnitude less than this of electrons. Hence they induce charge on the electrodes with hundred nanoseconds of duration. The method used to calculate the charge induced on an electrode is by using the Shockley-Ramo theorem and the concept of the weighting field. In the case of a charge q moving with a drift velocity u d rift the instantaneous current induced at a given electrode will be : i(t) = qu drift E w (1.5) where E w is the weighting field. 1. Gaseous Detectors The gaseous detectors have been employed and operated in various applications and experiments with every successful results over the last century. The differences between various types of gas counters with respect to their operation voltage is illustrated in figure 3. The number of ions pairs, equivalent to the detected pulse amplitude, is plotted as a function of the electric field for two different types of radiation.

17 3 1. Gaseous Detectors Figure 3: The different regions of gaseous detectors operation with respect to the applied voltage. 1. Recombination Region : in this region the electric fields are very low so the separation of the primary electron-ions pairs is not reliable and a fraction of the charge-pairs recombine with result we dont get any current.. Ion Chamber Region : in this region the voltage is enough to get electrons but cannot observe Avalanche and so the multiplication of the electrons. 3. Proportional Counting Region : in this region we can get the multiplication of proportional of applied voltage. 4. Limited of proportionality : at higher amplification the amount of charged ion in the vicinity of the anode increases and their space charge reduces the electric field by following electrons. We can see that the curve goes up so it does don t use by the detector. 5. Geiger Region : due to high voltage we have new Avalanches until the electric field to decreased so that to stop the multiplication and so we observe a constant value of multiplication. 6. Discharge Region : in this region the voltage is so high so we can observe a transition in the electric field with or without ionization and so it cannot be used by the detector.

18 1. Gaseous Detectors Micromegas Micromegas or Micro Mesh Gaseous Structure is a very assymetric double structure two stage parallel plate detector. Like the others gaseous detectors can detect charged and neutral particles. The difference with the others detector is that its two distinguished regions are no longer separated by a plane of wires but by a micromesh. Figure 4: Micromegas A MM consists of the following components : 1. anode electrode. Anode strips of gold-coated copper of 15 µm, with µm pitch, are printed on a 1 mm substrate. The thickness of the copper strip was 5 µm. Thinner strips were obtained by vacuum deposition. These allow a substantial reduction of the inter strip capacitance. Both metal-deposition techniques can be applied on a 5 µm thick Kapton substrate, whenever a reduction of the material of the detector is required. The strips were grounded through low-noise charge pre-amplifiers of high gain (4V/pC).. quartz fibres of 75 µm, with mm pitch, were stretched and glued on a G1 frame. The quartz frame was then mounted on the strip surface, defining a precise(%) gap. Thicker(14 and 3 µm) quartz spacers were also utilized during our tests. 3. the micromesh. In figure is a photograph of the micromesh obtained with a microscope. It is a metallic grid, 3 µm thick, with 17 µm openings every 5 µm. It is made of nickel, using the electroforming technique, which is flexible and exhibits as high degree of fidelity of the electroposited layer. 4. the conversion-drift electric field was defined by applying negative voltages on the micromsh (HV) and a slightly higher voltage on a second electrode (HV1), spaced by 3 mm in order to define a conversion-drift space. It was made by a standard nickel mesh, 1 µm thick, having 8% transparency, in order to allow a efficient penetration of the various radioactive sources used for the test and fixed on the top of the gross mesh. For the final detector, thin aluminized mylar can be used to define electrode HV1 and ensure at the same time the required gas tightness of the chamber. 5. the gas volume. The various elements of the parallel-plate structure were placed in a tight stainless steel vessel flushed by a standard gas mixture of Ar + 1% CH4 at atmospheric pressure. A metallic holder was mounted on top of the parallel plate chamber to support the radioactive source and a stainless steel collimator 1 mm thick with a mm hole. The metallic source holder can move horizontally and allow a rough scan of the active surface of the detector. The Micromegas detector can be separated on two region by the electric field. drift region multiplication region In the drift region we have the first ionization and due to not too strong electric field (1 5 kv/cm) the electrons are heading to the multiplication region without a a strong Avalanche. In the multiplication region the electric field is more stronger ( 1 kv/cm) and the result is a strong Avalanche and there is a strong signal on anode.

19 5 1. Gaseous Detectors Figure 5: Two areas of electric field Signal Formation When charges are moving in front of a conductor, a proportional charge is induced on the conductor. The Micromegas geometry, as shown in figure 4, contains the amplification gap, where the electric field is up to 5 kv/cm. This gives rise to an avalanche effect of ion-electron pairs being created due to ionizations. To simplify for this geometry for a charge q, the induced current is : I a = q E Au V a (1.6) with E A the electric field at position of the charge, u the velocity of the charge and V a the potential of the strips. Taking all the charges into account as a current density J(x,t) the induced current take the form : I a = 1 V a J(x, t) E(x, t)d 3 x (1.7) To find the total signal formation, we need to find the evolution of charge over time. The charge multiplication charge depends on the (first) Townsend coefficient a which in general is a function of the electric field. For n the number of one type of charged particles at a certain point,then their increase at a nearby point along the path of the moving charge will be dn = andr, where dr is the distance between the two points. To find the increase of charge we have to integrate the above formula. So : I(t) = q aβe aβupt u p V a E z (z)e aβz dz (1.8) where q the initial charge (one e ), u p /n the velocity of the ions/electrons, E z the z-component of the electric field. The derived current as a function of time is drawn in figure 6.

20 1. Gaseous Detectors 6 Figure 6: Total auxiliary current as a function of time. The peak at the origin lasts for less than ns is due primarily to the electrons movement while after ns the distribution id due to the ion drifting towards the mesh Resistive strips Micromegas Despite the excellent characteristics of the Micromegas module and the promising industrial bulk fabrication procedure, the very thin amplification region along with the finely sculpted readout structure makes them particularly vulnerable to discharges(sparks).sparks occur when the electron avalanche population goes beyond 1 6.Sparks may damage the detector and readout electronics and/or lead to large dead times as result of HV breakdown. To find a solution for this problem we create bulk-micromegas chambers spark resistant while mainting their ability to measure with excellent minimum-ionizing particles in high-rated environments. (a) view along the strip direction (b) side view,orthogonal to the strip direction Figure 7: Sketch of the detector principle(not the scale), illustrating the resistive protection theme In the figures 7 we see two orthogonal side views of the chamber. It is a bulk-micromegas structure built on top of a printed circuit board with 18 µm thick Cu readout strips covered by a resistive protection layer. The protection consists a thin layer of insulator on top of which strips of resistive paste (with a resistivity of a few ΜΩare deposited. Geometrically, the resistive strips match the pattern of the readout strips. They both are 15 µm wide and 8 µm long, their strip pitch is 5 µm. The resistive strips are 64 µm thick; the 1 µm wide gaps between neighboring strips are filled with insulator. The resistive strips are connected at one end to the detector ground through a 15 5 ΜΩ resistor, see below. We opted for resistive strips rather than a continuous resistive layer for two reasons: i) to avoid charge spreading across several readout strips, and ii) to keep the area affected by a discharge as small as possible. The Micromegas structure is built on top of the resistive strips. It employs a woven stainless steel mesh with 4 lines/inch and a wire thickness of 18 µm. The mesh is kept at a distance of 18 µm from the resistive strips by means of small pillars (4 µm diameter) made of the same photoimageable coverlay material that is used for the insulation layer. The pillars are arranged in a regular matrix with a distance between neighbouring pillars of.5 mm in x and y. The mesh covers an area of 1 1 mm. Above the amplification mesh, at a distance of 4 or 5 mm, another stainless steel mesh (35 textrmlines/inch, wire diameter: µm served as drift electrode. Its lateral dimensions are the same as for the amplification mesh. The chamber comprises 36 readout strips. The readout strips are left floating at one end. At the other end they are connected in groups of 7 strips to five 8-pin connectors. The remaining eight pins of each connector serve

21 7 1. Gaseous Detectors as grounding points. The detector housing consists of a mm high aluminum frame, mounted on top of the readout board and sealed by an O-ring, and a cover plate (again sealed by an O-ring) with some opening windows, made of 5 µm thick Kapton foil.

22 . Electric fields,weighting fields and signals in detectors including resistive materials 8 Electric fields,weighting fields and signals in detectors including resistive materials In this section we discuss the electric fields and the signals in detectors that represent parallel plate geometries with segmented readout like GEM s,micromegas,rpc s. In these detectors, the charges generated inside the sensor volume act as a source of the signal..1 Potential of a point charge centered at the origin Figure 8: A point charge Q on the boundary between two dielectric layers. We first investigate the electric field of a point charge in a two layer geometry. We assume that the two layers have thickness of b and g with constant dielectric permittivity of ɛ 1 and ɛ, surrounded by grounded metal plates. At r=,z=(the boundary between those two layers) we put a charge Q. We will use cylindrical coordinates due to problem s rotational symmetry. Because we don t have charges on the two layers we will use the Laplace equation. For r= the coefficients Y (kr) are zero so the general solution for the two areas will be : ϕ 1 (r, z) = 1 π ϕ (r, z) = 1 π J (kr)[a 1 (k)e kz + B 1 (k)e kz ]dk b < z < (.1) J (kr)[a (k)e kz + B (k)e kz ]dk < z < g (.) Boundary Conditions Because we have ground at z=g, z=-b we have the conditions ϕ 1 (r, b) = ϕ (r, g) = : A 1 e kb + B 1 e kb = (.3) A e kg + B e kg = (.4) At z=, the barrier between the two layers, we assume a surface charge density q(r). From Gauss Law for a medium inhomogeneous perimittivity we derive that passing through an infinitely thin sheet of charge with a surface charge density q(r), the pontential is continuous so ϕ 1 (r, ) = ϕ (r, ) which gives : A 1 + B 1 = A + B (.5) and the ɛe component perpendicular to the sheet "jumps" by q(r) : with q(r) = qδ(r) 1 π r 1 π ɛ 1 ϕ 1 (r, z) z ϕ (r, z) z= ɛ z= = q(r) z J (kr)[(kɛ 1 A 1 kɛ 1 B 1 ) (kɛ A kɛ B )]dk = qδ(r) 1 π r

23 9. Electric fields,weighting fields and signals in detectors including resistive materials by multiplying both sides with rj (k r) and integrate from to infinity : 1 π rj (kr)j (k r)dr [(kɛ 1 A 1 kɛ 1 B 1 ) (kɛ A kɛ B )]dk = rj (k r) qδ(r) 1 π r dr 1 π δ(k k )[ɛ 1 (A 1 B 1 ) ɛ (A B )]dk = Q π ɛ 1 (A 1 B 1 ) ɛ (A B ) = Q (.6) From (.3),(.4) : B 1 = A 1 e kb B = A e kg By using them on (.5) : A 1 ( 1 e kb ) = A ( 1 e kg ) A = 1 e kb 1 e kg And so the (.6) : ɛ 1 A 1 e kb cosh (kb) ɛ A 1 e kb ( e kb e kb) e kg (e kg e kg ) ekg ( e kg + e kg) = Q A 1 = Qsinh (kg) e kb (ɛ 1 cosh (kb) sinh (kg) + ɛ sinh (kb) cosh (kg)) we set D (k) = 4 (ɛ 1 cosh (kb) sinh (kg) + ɛ sinh (kb) cosh (kg)) and so A 1 = B 1 = A = B = Qsinh (kg) ekb D (k) Qsinh (kg) e kb D (k) Qsinh (kb) e kg D (k) Qsinh (kb) ekg D (k) And so the solution read as : φ 1 (r, z) = Q π J (kr) 4sinh (gk) sinh (k (b + z)) dk b < z < (.7) D(k) φ (r, z) = Q π J (kr) 4 sinh(bk)sinh((k(g z)) dk < z < g (.8) D(k)

24 . Electric fields,weighting fields and signals in detectors including resistive materials 1 Those integrals cannot be expressed in closed form, so we have to find some techniques to express the result as an infinite series. 4 sinh(gk) sinh(k(b+z)) We will take the integral quantity for φ 1 : D(k) and add and remove the quantity ekz ɛ 1+ɛ : We know that By doing that φ 1 take the form : e kz 4 sinh(gk) sinh(k(b + z)) + ɛ 1 + ɛ D(k) ekz ɛ 1 + ɛ = ekz ɛ 1 + ɛ + f 1 (k, z) 1 r + z = J (kr)e kz dk (.9) Q 1 φ 1 (r, z) = π(ɛ 1 + ɛ ) r + z + Q π J (kr)f 1 (k, z)dk (.1) Now the solution is the combination of the solution for the potential of a charge Q on the boundary of two infinite half-spaces of permittivity ɛ 1 and ɛ with a correction term. For the correction term, for large values of k the quantity f 1 (k, z) take the form : As result the potential for b < z < : f 1 (k, z) = e k(b+z) ɛ 1 + ɛ ɛ e k(g z) ɛ 1 + ɛ (.11) Q 1 φ 1 (r, z) = π(ɛ 1 + ɛ ) r + z Q Q π(ɛ 1 + ɛ ) r + (b + z) Qɛ 1 π(ɛ 1 + ɛ ) r + (g z) + Q π J (kr)f (k, z)dk The two new terms correspond to two "mirror" charges, one with value Q at z = b that is reflected at the grounded plate at z = b and one with value ɛ Q at z = g that is reflected at the grounded plate at z = g. If we want to find the φ is the same process and result is reversed. (.1). Dicharges on a resistive MICROMEGA From the previous subsection we know that the solution for a two layer problem with a charge Q on the boundary is : φ 1 (r, z) = Q π J (kr) 4 sinh(gk) sinh(k(b + z)) dk b < z < D(k) φ (r, z) = Q π J (kr) 4 sinh(bk) sinh(k(g z)) dk D(k) < z < g The electric field E z at r = and z = g is then E z = Q π k sinh(bk) dk (.13) D(k) We consider the following parameter for the resistive MICROMEGA : The distance between the mesh and the resistive strips(amplification gap) is 18 µm. Insulator between the resistive strips and the readout strips is 64 µm. We assume the permittivity of the insulating layer to be ɛ r = 5. The normal operation voltage is 5 V. Applying the 5 V between the resistive strips and the mesh gives a field of 5 V/18 µm = 39kV/cm. In case of a discharge there is a charge flowing from the mesh to the surface of the resistive layer over a short time. During this time the charge does not diffuse on the resistive layer, so we simply have a point charge accumulating at r = and z =. We assume now that the discharge stop when the electric field E z (due to the point charge) at the surface of the mesh equals the applied electric field.

25 11. Electric fields,weighting fields and signals in detectors including resistive materials With those parameters if we calculate the integral with the help of the Mathematica is gives as : ksinh(bk) dk = D(k) so to find the charge Q: E z = Q πɛ k sinh(bk) dk D(k) Q = πɛ E z Q = 14.8 pc This would be the approximate maximum charge in spark. There is no RC description of this situation. The charge has not yet started to diffuse and there is no equivalent capacitance anywhere. There is simply charge sitting on the surface of the resistive layer, producing and electric field that is counter-acting the applied field. If we assume a MICROMEGA of A = 1 1 cm size without metallic readout electrode and an avalance gap g, the total charge stored in the capacitor is A Q = ɛ nc g which is larger compare to the resistive layer case and this amount of charge will enter the amplifier. Now we will try to change the applied voltage(5 6 V ) to see how the charge Q will change. Table 1: Charge on Mesh for different values of Voltage. Applied Voltage(V) Electric Field E z Charge Q(pC) (kv/cm) With the results of the table we will create a figure with the Q and the applied voltage. Figure 9: Charge Q for different values of voltage. We see that there is a linear increase to the charge with the steadily increase of the voltage.

26 . Electric fields,weighting fields and signals in detectors including resistive materials 1 Table : Charge on Mesh for different values of g. g(µm) value of the integral Charge(pc) Now we will create the figure to see the relation between the g and and Q. We see again a linear increase to the charge with the steadily increase of the avalanche gap g. Figure 1: Values of charge for different values of distance between mesh and the resistive strips for applied voltage 57 V.3 Potential of a point charge in a geometry grounded on a rectangle For this case the geometry is grounded at x =, a and y =, b and the charge is placed at position x, y. Figure 11: A point charge Q in an empty condenser. We have to solve the Laplace equation in Cartesian coordinates. The general solution is : Φ 1 (x, y, z) = sin n=1 m=1 ( ) lxπ sin a ( myπ b ) [E 1 (k l m)e k lmz + F 1 (k l m)e k lmz ] (.14)

27 13. Electric fields,weighting fields and signals in detectors including resistive materials Boundary Conditions Φ 1 = Φ E 1 + F 1 = E + F z = (.15) Φ 1 = E 1 e kb + F 1 e kb = z = b (.16) Φ = E e kg + F e kg = z = g (.17) (.18) ɛ 1 [ sin n=1 m=1 and by integrate : b a ɛ 1 Φ 1 (r, z) z ( ) lxπ sin a ( l xπ sin a ) sin b a ( myπ b ( m yπ b Φ 1 (r, z) z= ɛ z= = Qδ(x x )δ(y y ) (.18) z ) (ke 1 kf 1 )] ɛ [ sin n=1 m=1 ( l ) ( xπ m ) yπ (.18) sin sin a b ) ( ) lxπ sin sin a ( myπ ( l xπ Qδ(x x )δ(y y ) sin a b ( ) lxπ sin a ( myπ ) [ke 1 kf 1 ) (ke kf )]dxdy = ) sin ( m yπ b b ) dxdy ) (ke kf )] = Qδ(x x )δ(y y ) for l = l and m = m : a b [(ke 1 kf 1 ) (ke kf )] = Qsin ( lx π ) ( a sin myπ ) b [ke 1 kf 1 ) (ke kf )] = 4Q Qsin ( ( lxπ a ) ( sin myπ ) b kab From (.16) and (.17) : So F = E 1 e kb e kg sinh(kb) sinh(kg) (.16) F 1 = E 1 e kb (.17) F = E e kg (.15) E 1 E 1 e kb = E E e kg E 1 e kb (e kb e kb ) = E e kb (e kg e kg ) E = E 1 e kb kg sinh(kb) e sinh(kg) (.18) ɛ 1 (E 1 + E 1 e kb ) ɛ ( E 1 e kb kg sinh(kb) e sinh(kg) E 1e kb e kg ) sinh(kb) sin ) = 4Q sinh(kg) ɛ 1 e kb E 1 (e kb + e kb ) + ɛ F 1 sinh(kb) sinh(kg) (e kg + e kg ) = 4 Q sin ( lxπ a ( lxπ a ) ( sin myπ ) b kab ) ( sin myπ ) b kab

28 . Electric fields,weighting fields and signals in detectors including resistive materials 14 From.1 : ɛ 1 e kb kb sinh(kb) sin E 1 cosh(kb) + ɛ E 1 e cosh(kg) = 4Q sinh(kg) E 1 = 4 Qsin ( lx π a ) ( sin myπ ) b kab ( lxπ a ) ( sin myπ ) b kab sinh(kg)e kb (ɛ 1 cosh(kb) sinh(kg) + ɛ sinh(kb) cosh(kg)) E 1 = Q sinh(kg)ekb D(k) 4 sin( lxπ a myπ ) sin( kab b ) = 4A 1 sin( lxπ myπ a ) sin( kab b ) (.19) F 1 = 4A 1sin ( lx π a kab ) ( sin myπ ) b e kb Q sinh(kg)ekb = 4 D(k) 4A 1 sin ( lx π a kab ) ( sin myπ ) b e kb F 1 = 4B 1 sin ( lx π a kab ) ( sin myπ ) b (.) So : Φ 1 (x, y, z) = sin n=1 m=1 Φ 1 (x, y, z) = ( ) lxπ sin a n=1 m=1 4 sin ( myπ b ) [ 4A 1 sin ( lx π a ( ) lxπ sin a ( myπ b kab ) ( sin myπ ) b e kz + 4B 1 sin ( lx π a kab ) ( sin myπ ) b ) ( sin lxπ ) ( a sin myπ ) b [A 1 e kz + B 1 e kz ] kab e kz ] k x = lπ a dk x = π a dl dl = a π dk x, dm = b π dk y Φ 1 (x, y, z) = 4 π [cos[k x (x x )] cos[k x (x+x )][cos[k y (y y )] cos[k y (y+y )] 1 k [A 1e kz +B 1 e kz ]dk (.1).4 Weighting Fields In this part we want to calculate the weighting field of a rectangular pad centred at x = y = with a width of w x and w y for the geometry of figure, which is infinitely extended and where the permittivity of both layers is equal to ɛ. We will use the previous solution and shift the coordinate system such there is a grounded plate at z = and z = g and the point charge at x,y,z.

29 15. Electric fields,weighting fields and signals in detectors including resistive materials Figure 1: rectangular readout pad We use the A 1 and B 1 from 3.1 and replace g = d z,b = z. D(k) = 4[ɛ 1 cosh(bk) sinh(gk) + ɛ sinh(bk) cosh(gk)] e bk + e bk e gk e gk D(k) = 4[ɛ 1 + ɛ e bk e bk e gk + e gk ] ɛ 1 = ɛ = ɛ ɛ [(e bk + e bk )(e gk + e gk ) + (e bk e bk )(e gk + e gk )] = aɛ [e k(b+g) e k(b+g) ] D(k) = 4ɛ sinh[k(b + g)] = 4ɛ sinh[k(z + d z )] = 4ɛ sinh[kd] (.) A 1 = Q sinh(k(d z )) e kz (.3) 4ɛ sinh(kd) B 1 = Q sinh(k(d z )) e kz (.4) 4ɛ sinh(kd) So to find the potential to area 1 : Φ 1 (x, y, z) = 4 π [cosk x (x x ) cosk x (x+x )][cosk y (y y ) cosk y (y+y )] 1 k Q sinh(k(d z )) [e kz e kz +e kz e kz ]dk x dk y 4ɛ sinh(kd) Φ 1 (x, y, z) = Q π ɛ [cosk x (x x ) cosk x (x+x )][cosk y (y y ) cosk y (y+y )] 1 k sinh[k(d z )]sinh[k(z + z )] dk x dk y sinh(kd) Φ 1 (x, y, z) = Q π ɛ [sin(k x x) sin(k x x )sin(k y y) sin(k y y )] 1 k sinh[k(d z )] sinh[k(z + z )] dk x dk y (.5) sinh(kd) Φ is given if we exchange z with z In this rectangular pad an charge Q i nd is induced : Q i nd(x, y, z ) = +wx wx + wy wy ɛ ϕ 1 z z=dxdy (.6)

30 . Electric fields,weighting fields and signals in detectors including resistive materials 16 φ 1 z = Q π ɛ [sin(k x x) sin(k x x ) sin(k y y) sin(k y y )] 1 k sinh[k(d z )](ke kz e kz + ke kz e kz ) dk x dk y sinh(kd) φ 1 z z= = Q π ɛ [sin(k x x) sin(k x x ) sin(k y y) sin(k y y )] sinh[k(d z )] cosh(kz ) sinh(kd) dk x dk y From the reciprocity theorem we know that Q i nd = Q/V w ϕ w (x, y, z ) where ϕ w is the potential at x,y,z in case is removed and the pad is put to potential V w. So : φ w (x, y, z ) = V wq i nd Q φ w (x, y, z ) = 4V w π cos(k x x ) sin( kxwx ) cos(k y y ) sin( kywy ) cosh(kz )sinh[z(d z )] dxdy k x k y sinh(kd) φ w (x, y, z ) = 4V w π.5 N-Layer geometry cos(k x x) sin( kxwx ) cos(k y y) sin( kywy ) cosh(kz) sinh[k(d z)] dxdy (.7) k x k y sinh(kd) Figure 13: A geometry on N dielectric layers enclosed by grounded metallic plates. On the boundary between two layers at r = there are point charges Q. We will approach the geometry in figure 13. We assume N dielectric layers ranging from z n 1 < z < z n of constant permittivity ɛ n. On the boundaries at z = z n, there are charges Q n. At z = z and z = z N there are grounded metal plates. We define a characteristic function f n (k, z) for each layer as f n (k, z) = A n e kz + B n e kz n = 1...N (.8) With this characteristic function we can find the solution for different geometries : For an infinitely extended geometry with the chargers at position x,y in Cartesian coordinates is given by

31 17. Electric fields,weighting fields and signals in detectors including resistive materials : with k = k x + k y Φ n (x, y, z) = 1 π sin(k x x) sin(k x x ) sin(k y y) sin(k y y ) f n(k, z) dk x dk y (.9) k For the case where the geometry is grounded on a rectangle at x =, a and y =, b the solution is : Φ n (x, y, z) = 4 ab sin l=1 m=1 ( ) lxπ sin a ( ) lx π ( myπ ) ( my π ) fn (k ml, z) sin sin a b b k (.3) l with k = π a + m b For the case where the geometry is grounded on a rectangle at x =, a and insulated at y =, b the solution is : Φ n (x, y, z) = 4 ab sin l=1 m= ( ) lxπ sin a ( lx π a ) cos ( myπ b ) cos ( my π b ) (1 δ m/) f n(k ml, z) k (.31) The N coefficients A n (k) and B n (k) are defined by the two conditions at the grounded plated and at the (N-1) conditions at the N-1 dielectric interfaces : A 1 e kz + B 1 e kz = A N e kz N + B N e kz N = A n e kzn + B n e kzn = A n+1 e kzn + B n+1 e kzn ɛ n (A n e kzn B n e kzn ) ɛ n+1 (A n+1 e kzn + B n+1 e kzn ) = Q n From these equations we will approach a general solution for the different forms of RPCS we are going to discuss in this section. We will start with a 3-layer geometry. Figure 14: A geometry with 3 dielectric layers. Boundary Conditions

32 . Electric fields,weighting fields and signals in detectors including resistive materials 18 A 1 e kz + B 1 e kz = (.3) (.33) A 1 E kz1 + B 1 e kz1 = A e kz1 + B e kz1 (.34) (.35) f 1 ɛ 1 z ɛ f z = Q 1 (.36) (.37) ɛ 1 A 1 e kz1 ɛ 1 B 1 e kz1 ɛ A e kz1 + ɛ B e kz1 = Q 1 (.38) (.39) A E kz + B e kz = A 3 e kz + B 3 e kz (.4) (.41) f ɛ z ɛ f 3 3 z = Q (.4) (.43) ɛ A e kz ɛ B e kz ɛ 3 A 3 e kz + ɛ 3 B 3 e kz = Q (.44) (.45) A 3 e kz3 + B 3 e kz3 = (.46) For these equation we create the M matrix for this 3-layer geometry. The equation to solve is then : M a = b (.47) e kz e kz e kz1 e kz1 e kz1 e kz1 M = ɛ 1 e kz1 ɛ 1 e kz1 ɛ e kz1 ɛ e kz1 e kz e kz e kz e kz ɛ e kz ɛ e kz ɛ 3 e kz ɛ 3 e kz 1 e kz3 e kz3 (.48) a = (A1, B1, A, B, A3, B3) T b = (,, Q1,, Q, ) T Last we will approach a generalization of (.1) from.1. We will use a example for the infinitely extended geometry in cylindrical coordinates with the charges centred at r = and have n = 1,...N. Q n 1 1 φ n (r, z) = π(ɛ n 1 + ɛ n ) r + (z z n 1 ) + Q n 1 1 π(ɛ n + ɛ n+1 ) r + (z z n ) π J (kr)[a n (k)e kz +B n (k)e kz Q n 1 e k(z zn 1) Q n e k(zn z) dk (.49) ɛ n 1 + ɛ n ɛ n + ɛ n+1.6 Single Layer RPC We will now approach a geometry with 3 layers(figure 15), a single gap RPC with one resistive layer. We will start by finding the coefficients.

33 19. Electric fields,weighting fields and signals in detectors including resistive materials Figure 15: Geometry with three layers and one point charge representing a single gap RPC. A 1 e kb + B 1 e kb = (.5) A 1 + B 1 = A + B (.51) ɛ 1 ka 1 ɛ 1 kb 1 ɛ ka + ɛ kb = (.5) A e kz + B e kz = A 3 e kz + B 3 e kz (.53) ɛ ke kz ɛ kb e kz ɛ 3 ka 3 e kz + ɛ 3 kb 3 e kz = Q (.54) A 3 e kg + B 3 e kg = (.55) From (.51) : B 1 = A 1 e kb By using (.51) on (.5): A 1 A 1 e kb = A + B We multiply (.5) with ɛ k and add with (.53) and we have : A 1 k(ɛ 1 + ɛ ) + kb 1 (ɛ ɛ 1 ) ɛ ka = ɛ ka = A 1 k(ɛ + ɛ ɛ r A 1 ke kb (ɛ ɛ ɛ r )) A = A 1 [(1 + ɛ r) e kb (1 ɛ r )] with ɛ 1 = ɛ ɛ r, ɛ = ɛ, ɛ 3 = ɛ By returning to (.5) : A 1 A 1 e kb = A 1 [(1 + ɛ r) e kb (1 ɛ r ) + B ] B = A 1 [1 e kb 1 (1 + ɛ r) + e kb e kb ɛ r ]

34 . Electric fields,weighting fields and signals in detectors including resistive materials B = A 1 [(1 ɛ r) e kb (1 + ɛ r )] From (.55): B 3 = A 3 e kg Now we add them together the relations (.53) and (.54) : By using B 3 from (.55) and A from above : A e kz B e kz A 3 e kz + B 3 e kz = Q ɛ k and so A 3 = A 1 [(1 + ɛ r) e kb (1 ɛ r )] Q e kz kɛ Now its important to find A 1. From (.54): B 3 = A 1 [(1 + ɛ r) e kb (1 ɛ r )]e kg + Q e kz kɛ e kg ɛ ka e kz ɛ kb e kz ɛ ka 3 e kz + ɛ kb 3 e kz = Q A 1 e kz [1 ɛ r ) + (1 + ɛ r )e kg e kb (1 + ɛ r ) e kb e kg (1 ɛ r ) = Q ɛ k Q ɛ e kz e kg The relation inside the bracket can become : e kg (e kg + ekg) e kb (1 + e kg ) + ɛ r ( 1 + e ekg ) + ɛ r (e kb e kg e kb) (e kg + e kg )e kg (e kb e kb + ɛ r e kg e kb (e kg e kg (e kb + e kb )) 4 cosh(kg) sinh(kb)e kg e kb + 4ɛ r sinh(kg) cosh(kb)e kg e kb 4e kg e kb D(k) withd(k) = cosh(kg) sinh(kb)e kg e kb + ɛ r sinh(kg) cosh(kb) Now returning to the previous relation to find A 1 : A 1 e kz 4e kg e kb D(k) A 1 = Qekb 4ɛ D(k) (ek(g z) e k(g z ) A 1 = Qekb sinh(k(g z)) (.56) ɛ D(k)

35 1. Electric fields,weighting fields and signals in detectors including resistive materials And so to find the others variables : A = A 1 [(1 + ɛ r) e kb (1 ɛ r )] A = Q 4ɛ D(k) sinh(k(g z ))[sinh(kb) + ɛ r cosh(kb)] A = Q ɛ D(k) sinh(k(g z ))[sinh(kb) + ɛ r cosh(kb)] (.57) B = Q ɛ D(k) sinh(k(g z ))[sinh(kb) ɛ r cosh(kb)] (.58) A 3 = Q sinh(k(g z )) (sinh(kb) + ɛ r cosh(kb)) Qe kz ɛ D(k) ɛ k (.59) B 3 = Qe kz e g ɛ k Q ɛ kd(k) sinh(k(g z ))[sinh(kb) + ɛ r cosh(kb)e kg ] (.6) As we told before the solution on different areas are on form of : f i = A i e kz + B i e kz We will show the potential on areas and 3,that they have a charge Q between them. f = A e kz + B e kz f = Q ɛ D(k) sinh(k(g z ))[sinh(kb)e kz + e r cosh(kb)e kz + sinh(kb)e kz e r cosh(kb)e kz ] f = Q ɛ D(k) sinh(k(g z ))[sinh(kb)(e kz + e kz ) + e r cosh(kb)(e kz e kz )] f = Q ɛ D(k) sinh(k(g z ))[sinh(kb) cosh(kz) + e r cosh(kb) sinh(kz)] (.61) f 3 = A 3 e kz + B 3 e kz f 3 = Q ɛ D(k) sinh(k(g z ))[sinh(kb) + e r cosh(kb)]e kz Q ɛ k e kz e kz + Q ɛ k e kz e kg e kz ] f 3 = Q ɛ D(k) sinh(k(g z))[sinh(kb) cosh(kz ) + e r cosh(kb) sinh(kz ) (.6) If we assume that the charge are on r = the the voltages for these areas are in form of : ϕ (k, z) = 1 π From (.49) : J (kr)f (k, z)dk ϕ 3 (k, z) = 1 π J (kr)f 3 (k, z)dk φ (r, z) = Q 1 1 4π(ɛ r + (z z) π J (kr)[f (k, z) Q ɛ e k(z z) ]dk (.63)

36 . Electric fields,weighting fields and signals in detectors including resistive materials φ 3 (r, z) = Q 4π(ɛ 1 r + (z z ) 1 π J (kr)[f 3 (k, z) Q ɛ e k(z z) ]dk (.64) The expressions represent a point charge Q in free space together with a term that accounts for the presence of the dielectric layer and the grounded plate, which is more suited for numerical evaluation..6.1 Weighting fields To find the weighting potential for a readout pad,readout strip and the full electrode we use (.9) : Φ 1 (x, y, z) = 1 π sin(k x x) sin(k x x ) sin(k y y) sin(k y y ) Q sinh[k(g z )] sinh[k(z + b)] dk x dk y (.65) ɛ kd(k) The Q ind for a readout pad is : Q ind = +wx wx + wy wy ɛ r + ɛ ϕ 1 z z=dxdy We know that : Q ind = 4Qɛ r π and by the reciprocity theorem : φ w (x, y, z) = 4V wɛ r π ( sin ( sin k x w x k x w x ) ( sin k y w y ) Q ind = Qφ w(x, y, z ) V w ) ( sin k y w y sin(k x x ) sin(k y y ) sinh[k(g z )] dk x dk y k x k y D(k) ) sin(k x x ) sin(k y y ) sinh[k(g z)] dk x dk y (.66) k x k y D(k) For a readout strip w y with k y = sy w y φ w (x, z) = 4V wɛ r π ( sin k x w x φ w (x, z) = 4V wɛ r π ( ) ) syy sinh[k(g z)] cos w y sin(sy) sin(k x x ) k x k y D(k) φ w (x, y, z ) = V wɛ r π ( sin For the full electrode w x w y k = : k x w x ( sin ) sin(k x x ) k x w x ) sin(k x x ) sinh[k(g z)] π k x k y D(k) dk x sy ds y dk x w w y y sinh[k(g z)] dk x (.67) kd(k) sinh[k(g z)] kd(k) because for small values of k sinh(x) x and cosh(x) 1 And so for the weighting potential : = k(g z) kb + kgɛ r φ w (z) = ɛ rv w (g z) b + gɛ r (.68)

37 3. Electric fields,weighting fields and signals in detectors including resistive materials Figure 16: Weighting field E z at position z = g/ for b = 4g and w x = g. The three curves represent ɛ r = 1(bottom),ɛ r = 8(middle),ɛ r = (top). In figure 17 we can see the a weighting field for a strip electrode of width w x and infinity extension Figure 17: Weighting field for a strip electrode of width w x and infinity extension. We first assume the geometry to represent a single layer RPC with a gas gap of g =.5 mm and a resistive layer of dielectric permittivity ɛ r and thickness b = 1 mm. We assume a very wide readout strips width w x = 5 mm and we find for the z-component of the weighting field in the center of the gas gap(z =.15 mm). as shown in figure 16. The three curves represent dielectric permittivities of ɛ r = 1(bottom), 8(middile) (top). The strip extends between 1 < x/g < 1 and the value at x/g = 1 is therefore half of the peak as required by symmetry for a wide readout strip. The value in the center of the strip is close to the one from.68 for the infinitely wide strip and it is clear from this expression that a higher dielectric permittivity of the resistive plate will increase the weighting field and therefore the induced signal. The value ɛ r = 8 which is typical for glass and bakelite used in RPC s gives a shape that is already close to the one for an arbitrarily large permittivity..6. Effect of Resistivity Figure 18: A geometry with three layers and one point charge.

38 . Electric fields,weighting fields and signals in detectors including resistive materials 4 Now we will calculate what happens when we put a point charge Q on the surface of the resistive plate at t = as shown in figure 18 : for ɛ 1 = ɛ ɛ r + σ/s ɛ = ɛ + Q 1 = Q/s E 1 (r, z, s) = Q sπ E (r, z, s) = Q sπ sinh(gk) cosh(k(b + z)) kj (kr) dk (.69) ɛ [sinh(bk) cosh(gk) + (ɛ r + σ/(ɛ s) cosh(bk) sinh(gk) sinh(bk) cosh(k(g z)) kj (kr) dk (.7) ɛ [sinh(bk) cosh(gk) + (ɛ r + σ/(ɛ s) cosh(bk) sinh(gk) We want to know the stationary situation so : lim se(r, z, s) : s E 1 (r, z) = I σπ kj (kr) cosh(k(b + z)) dk (.71) cosh(bk) with Q = I /S E (r, z) = I σπ kj (kr) tanh(bk)cosh(k(g z)) dk (.7) sinh(gk) We see that E 1 does not depend on g but depends only on the thickness b of the resistive layer. This is evident from the fact that there is no DC current that can flow through the gas gap, so only the geometry of the resistive layer is relevant. The current density i (r) flowing into the grounded plate at z = b is related on the field on the surface of the grounded plate by i (r) = σe 1 (r, z = b) = I 1 ( yr ) b π J y dy (.73) b cos(y) with y = bk For small values of r we can insert the series expansion for J (x) and evaluate the integrals gives : 1 ( yr ) J y b cos(y) (r b ) ( r b )4... (.74) For large values of r : 1 J ( yr ) b y cos(y) π r/b e rπ b r b 1 (.75) Both those approximation of current are plotted in figure 19 and we see that for r/b > the exponential approximation describes the situation already to very high accuracy.

39 5. Electric fields,weighting fields and signals in detectors including resistive materials Figure 19: Current density i (r) at z = b. The blue curve represent the second order approximation of (.74), the green curve the fourth order approximation of (.74) and the yellow curve the approximation of (.75)..6.3 Surface resistivity (a) Genetal 3 layer geometry with a point charge Q. (b) A resistive plate with conductivity σ together with an thin layer of surfave resistivity RΩ /square and impressed current I Figure The glass or Bakelite might develop a conductive surface once the electric field is applied. formalism for 3 layer geometry like before with : We employ the ɛ 1 = ɛ ɛ r + σ/s ɛ = ɛ + 1 sz R ɛ 3 = ɛ Q 1 = Q = I s with z = b z 1 = z z 3 = g and we perform the z z 1 = With the previous process and by using lim s sf(k, z, s) : f 1 (k, z) = I [sinh[k(b + z)]] (.76) σ[cosh(kb) + k/(rσ) sinh(bk)] f 3 (k, z) = 1 σ I [sinh(kb) sinh[k(g z)]] (.77) sinh(kg)[cosh(kb) + k/(rσ) sinh(bk)] E 1 (r, z) = I σπ cosh[k(b + z)] kj (kr) dk (.78) cosh(kb) + k/(rσ) sinh(bk) E 3 (r, z) = I σπ sinh(kb) cosh[k(g z)] kj (kr) dk (.79) sinh(kb) + k/(rσ) sinh(bk)

40 . Electric fields,weighting fields and signals in detectors including resistive materials 6 And the current i (r) : with β = Rσb i (r) = σe 1 (r, z = b) = I 1 b π J ( yr ) b y cos(y) + y dy (.8) β sinh(y) In the limit of high resistivity R we recuperate the expression from the previous section without any resistive surface layer. i (r) = I b π β π e βr b βr b (.81) Comparing this with relation (.75) we see that the radial exponential decay of the current is not any more governed ny the characteristic length b/π by b/β..7 Single Thin Resistive Layer Now we want to study the fields of a single layer of surface resistivity R at z =,where we place a charge Q at r = at t =. We write Q(t) = QΘ(t) with Θ(t) the Heavyside step function. If we use the Laplace domain in take the form Q(s) = Q/s. (a) A resistive layer with surface resistance R. (b) 3 layer geometry In this case also we use a 3-layer geometry with variables : ɛ 1 = ɛ ɛ = ɛ + 1 sz R ɛ 3 = ɛ Q 1 = Q s Q = By taking the limits : z z 1 = z z 3 + The solution for the 3 areas are : f 1 = A 1 e kz + B 1 e kz f 1 = A 1 e kz f = A e kz + B e kz f 3 = A 3 e kz + B 3 e kz f 3 = B 3 e kz B 1 = and A 3 = because at the infinities the quantities f 1, f must be zero. Boundary conditions

41 7. Electric fields,weighting fields and signals in detectors including resistive materials f 1 = f A 1 = A + B z = z 1 = (.8) ( ɛ A 1 ɛ + 1 ) (A B ) = Q ɛ A 1 (ɛ + 1 ( )A + ɛ + 1 ) B = Q z = z 1 = (.83) srz srz srz A e kz + B e kz = B 3 e kz z = z (.84) ( ɛ + 1 ) ( A e kz ɛ + 1 ) B e kz + ɛ B 3 e kz3 = z = z (.85) srz srz We multiply (.8) with (ɛ + 1 srz ) and add it with (.83) : ( ɛ + 1 ) A 1 srz ( ɛ + 1 srz ) A = Q A = A 1 ɛ srz + 1 QsRz (srz ɛ + 1) (.86) B = A 1 A = A 1 A 1 ɛ srz + 1 QsRz (srz ɛ + 1) B = A 1 + QsRz (srz ɛ + 1) (.87) from (.84) by using A and B from above : B 3 = Now from all the above we have: A 1 (srz ɛ + 1) [(srz QsRz ɛ + 1)e kz + 1] (srz ɛ + 1) (ekz 1 ) (.88) A 1 = Q srz ɛ e kz + cosh(kz ) k(srz ɛ + 1) srz (ɛ srz + 1) (ɛ srz ) e kz + srz ɛ e kz + sinh(kz ) srz ɛ e kz + cosh(kz )srz A 1 = Q ɛ srz e kz (ɛ srz + 1) + sinh(kz ) A 1 = Q (srz ) ɛ e kz + e kz +e kz srz ɛ srz e kz (ɛ srz + 1) + ekz e kz (.89) If we use z : and if we use Q = Q s A 1 = QsR k(ɛ sr + k) A 1 = QR ɛ sr + k (.9) From (.86),(.87) : A = QR (ɛ sr + k) = B (.91)

42 . Electric fields,weighting fields and signals in detectors including resistive materials 8 With the knowledge of the 3 of 4 coefficients we can go now to (.88) and find the B 3 : B 3 = QR ɛ sr + k (.9) If we return to the solutions f 1,f,f 3 : f 1 = A 1 e kz f 1 = f = A e kz + B e kz f = QR ɛ sr + k ekz (.93) QR cosh(kz) for z = (.94) ɛ sr + k f 3 = B 3 e kz f 3 = QR ɛ sr + k e kz (.95) We will use the solution f 1 and f 3 for some cases of layers in the next parts. time domain : We prefer to use them in the f 1 = Q ɛ e k(vt z) f 3 = Q ɛ e k(vt+z) for v = 1 ɛ R.7.1 Infinitely extended resistive layer (a) A point charge placed at an infinitely extended resistive layer at t =. (b) The solution for the time dependent potential is equal to a point charge moving with velocity v along the z-axis. Figure 1 We will start from an infinitely extended layer. The charge Q will cause currents to flow inside the resistive layer that are "destroying" it. The solution for the potential is : φ 1 (k, z, t) = Q 4πɛ J (kr)e k(vt z) dk (.96) and from (.9) : φ 3 (k, z, t) = Q 4πɛ φ 1 (r, z, t) = φ 3 (r, z, t) = J (kr)e k(vt+z) dk (.97) Q 1 (.98) 4π(ɛ r + ( z + vt) Q 4π(ɛ 1 r + (z + vt) (.99)

43 9. Electric fields,weighting fields and signals in detectors including resistive materials The potential to the point charge placed on the infinitely extended resistive layer at t = is equal to the potential of a charge Q that is moving with velocity v = 1 ɛ R away from the layer along the z-axis. We can calculate the charge density q(r,t) on the resistive layer through Gauss law : q(r, t) = ɛ ϕ 1 z z= ɛ ϕ 3 z z= q(r, t) = Q π vt r + (v + t ) 3 (.1) Lastly we calculate the total current I(r) flowing radially through a circle of radius r : I(r) = rπ ϕ 1 E(r) = rπ R R r Qvr r= = r + (v t ) 3/ (.11).7. Resistive layer grounded on a rectangle Figure : A point charge placed on a resistive layer that is grounded on a rectangle. In this case the resistive layer a grounded boundary at x =, x = a and y =, y = b and place a charge Q at position x, y at t =. The potential is given by (.3). We assume that the currents pointing outside of the boundary, the currents flowing through 4 boundaries are : I 1 x(t) = 1 R I 1 y(t) = 1 R b ϕ1 x x=dy a ϕ1 y y=dy I x(t) = 1 R I y(t) = 1 R b ϕ1 x x=ady a ϕ1 y y=bdy with 1 R = ɛ v φ 1 = 4 ab sin l=1 m=1 ( ) lxπ sin a ( lx π b ) ( myπ ) sin sin a ( my π b ) f1 k with f 1 = Q ɛ e k(vt z) We will start with the I 1 x(t)

44 . Electric fields,weighting fields and signals in detectors including resistive materials 3 φ 1 = Q ɛ 4 ab sin l=1 m=1 φ 1 x x= = Q ɛ 4 ab ( ) lxπ sin a l=1 m=1 lπ a sin ( ) lx π ( myπ ) ( my π ) e k(vt z) sin sin b a b k ( ) lx π ( myπ ) ( my π ) e k(vt z) sin sin b a b k ɛ v Q ɛ 4 ab I 1x (t) = 4Qv a l=1 m=1 I 1 x(t) = 1 R b lπ a sin(lx π ) sin b l=1 m=1 By doing the same we have for the other currents : I x(t) = 4Qv a l=1 m=1 l m [1 ( 1)m ] sin l m ( 1)l [ 1 + ( 1) m ] sin φ 1 x x=dy = ( myπ ) ( my π sin a b ) e k(vt z) dy k ( ) lx π ( my π ) e k(vt z) sin b b k ( ) lx π ( my π ) e k(vt z) sin b b k (.1) (.13) I 1 y(t) = 4Qv b l=1 m=1 m l [1 ( 1)l ] sin ( ) lx π ( my π ) e k(vt z) sin b b k (.14) I y(t) = 4Qv b l=1 m=1 m l ( 1)m [ 1 + ( 1) l ] sin ( ) lx π ( my π ) e k(vt z) sin b b k (.15).7.3 Resistive layer grounded at ±a and insulated at ±b Figure 3: A point charge placed on a resistive layer that is grounded on at x = and x = a but insulated on the others border. In this case the resistive layer is grounded at x =, x = a and insulated at y =, y = b. The currents are only flowing into the grounded elements at x =, x = a. By using (.31) we can have : ) I 1 x(t) = 1 R I x(t) = 1 R For big large times both the expressions : ϕ 1 x x=dy = Q T π ϕ 1 x x=ady = Q T π sin ( x π a cosh ( t T ) cos((fracx πa) sin ( x π ) a cosh( t T ) + cos ( x π a (.16) ) (.17) I 1 x(t) = I x(t) Q ( T π sin x π ) e t T (.18) a

45 31. Electric fields,weighting fields and signals in detectors including resistive materials Both those two currents and the approach for large values can be observe on the next figure. We assume that we have a charge deposit at position x = a/4 Figure 4: Currents I 1x (t)(top) and I x (t)(bottom) from the of figure 3 for x = a/4. The straight line in the middle refers to the approximation from Resistive layer parallel to a grounded plane (a) A resistive layer with surface resistance R in presence of a ground layer at distance b (b) 3-layer geometry by performing the indicated limits of the expressions for z,z 3 Figure 5 In this part we want to study the fields and charges in a layer of surface resistivity R at z = where we place a charge Q. AT r = at t = in presence of a grounded plane at z = b as shown in figure 5. Like before we have ɛ 1 = ɛ ɛ = ɛ + 1 sz R ɛ 3 = ɛ Q 1 = Q s Q = By taking the limits : z = b z 1 = z z 3 + The solution for the 3 areas are : f 1 = A 1 e kz + B 1 e kz f 1 = A 1 e bk + A e kb f = A e kz + B e kz f 3 = A 3 e kz + B 3 e kz f 3 = B 3 e kz A 3 = because at the infinity the quantity f must be zero.

46 . Electric fields,weighting fields and signals in detectors including resistive materials 3 By using the previous process or with the help of some programs like Mathematica we find : A 1 = QRekb D(k) B 1 = QRe kb D(k) A 3 = B 3 = QR sinh(kb) D(k) with D(k) = k sinh(kb) + e kb ɛ Rs In the Laplace domain : A 1 = QRekb D(k) B 1 = QRe kb D(k) A 3 = B 3 = QR sinh(kb) D(k) (.19).8 Uniform currents on thin resistive layers In this part we discuss the potentials that are created on thin resistive layers from uniform charge deposition. In detectors like RPC s and Resistive Micromegas such resistive layers are used for application of the high voltage and for spark protection. The resistivity must be chosen small enough to ensure that potentials that are established on these layers due to charge-up are not influencing the applied electric fields responsible for the proper detector operation. If such detectors are in an environment of uniform particle irradiation the situation can be formulated by placing a uniform "externally impressed" current per unit area i on the resistive layer. Figure 6: Uniform current "impressed" on the resistive layer will result in a potential distribution that depends strongly on the boundary conditions. First we will investigate the geometry shown in figure 5. We have a charge dq at position x and y and after a time t is given by : dq(t) = i r dr dφ t and in Laplace domain dq(s) = i r dr dφ 1 s From.7 we replace Q/s by q(s) so : f 1 (k, z) = Ri r dr dφ k e kz f (k, z) = Ri r dr dφ e kz (.11) k In this geometry we have a rectangular. We replace r dr dφ by dx dy and perform integration a dx b dy on (.3) and so : φ 1 (x, y, z) = φ 3 (x, y, z) = abri 4 π 4 l=1 m=1 [1 ( 1) l ][1 ( 1) m ] sin( lxπ myπ a ) sin( l 3 mb/a + m 3 la/b b ) e k lmz (.111) This expression cannot be written in closed form. The peak can be found by setting dφ 1 /dx=,dφ 1 /dy =. It can be found at x = a/ and y = a/, which is also evident for symmetry on this geometry. The maximum potential on the resistive layer is then : φ max (a/, b/, z = ) 1 8 a b Ri 18 π 4 l=1 m=1 ( 1) l+m b (l 1) 3 (m 1) + a (m 1) 3 (l 1) (.11)

47 33. Electric fields,weighting fields and signals in detectors including resistive materials for square geometry (a = b) the sum evaluates to.59 so the peak voltage in the center is : φ max.74ri a =.74RI tot (.113).9 Uniform Currents on a resistive plate covered with a thin resistive layer (a) Geometry to define a single resistive layer of thickness d covered by a resistive surface layer of resistance R Ω/ Square. (b) Uniform currents of+i, i on the top and bottom of the surface Figure 7 In this part we want to investigate the potential drop across a rectangular resistive plate that is covered by a thin resistive layer, that is grounded on two sides. From before in this case we have : ) f 1 (k, z) = i σ f 5 (k, z) = i σ cosh ( kd cosh ( kd e kz sinh ( kz ) + k Rσ sinh ( kd e k(d z) sinh ( ) kz ) + k Rσ sinh ( kd ) (.114) ) (.115) We can see that f 5 (k, z) = f 1 (k, d z), which is due to the symmetry of the problem. For uniform illumination we proceed as before by talking the solution a grounded geometry at x = and x = b and insulated at y = and y = b and we write i dx dy and integrate over x and y, which gives : [1 ( 1) l ] sin ( ) lxπ a φ n (x, z) = a l π f n (lπ/a, z) (.116) l=1 The potential difference between the top and the bottom of the plate, is given by : V (x) = φ 1 (x, z = ) φ 5 (x, z = d) = 4ai σ The maximum potential found at x = a/ and evaluates at : V (a/) = fracai σ l=1 [1 ( 1) l ] sin ( ) lxπ a l π [1 ( 1) l ] ( 1) (l+1)/ l=1 l π For a infinitely long layer we expand the expression for small values of d/a : V (a/) fracai σ [1 ( 1) l ] ( 1) (l+1)/ l=1 l π cosh ( ldπ a ( ldπ a sinh ) + lπ sinh( ldπ a ) cosh( ldπ a ) + lπ arσ ) arσ sinh ( ldπ a ) (.117) sinh( ldπ a ) (.118) ldπ a = i σ d = V (.119) which is the expected expression for the voltage drop across s resistive plate. The effective resistance of a small

48 . Electric fields,weighting fields and signals in detectors including resistive materials 34 surface A id therefore given by : with R = d σa V = i σ d = i A σa d = I d σa = I R (.1) In case the plate resistivity is much larger than the surface resistivity we can neglect the first term in the denominator and the expression evaluates to : V (a/) = 1 4 a Ri := V 1 (.11) The effective resistance of a small surface A is : V 1 = 1 4 a Ri = 1 4A a RAi = 1 4A a RI = R 1 I (.1) with R 1 = 1 4A a R The transition between the two cases pf surface resistivity only and bulk resistivity only is therefore given when R 1 = R For R = R 1 R = 4D σa := R eff (.13) The expression for the potential difference across the plate can be written : V (x) = 4ai [1 ( 1) l ] sin ( ) ( lxπ a sinh ldπ ) a σ l π l=1 cosh ( ) ldπ R a + eff lπa R 4d sinh ( ) (.14) ldπ a and defining f = d/(a), the maximum potential V in the center of the resistive plate at x = a/ : The expression for the potential difference across the plate can be written : V (x) V = [1 ( 1) l ] ( 1) (l+1)/ l=1 lπ sinh cosh (lπf) + R eff R ( ) lfπ lπf lπ 8f sinh (lπf) (.15) As an approximate model one can assume that the current that is placed on the resistive plate will divide to the effective resistances R 1 and R, so the voltage drop should be given by V R R 1 R + R 1 I = V 1 + R eff R (.16) This expression is very close to the first term of the sum in (.15),(l = 1) in face of f 1. For practical RPC s applications with glass RPC s, the glass thickness is typically.4 mm and the length a is about a =.7cm, so a typical value is f =.1. The figure 8 shows both those expressions and we see that the expression works quite well.

49 35. Electric fields,weighting fields and signals in detectors including resistive materials Figure 8: Voltage across the center of the resistive plate for a value of f = d/(a) =.1. The dots refer to the exact formula (.15), the curved line corresponds to the approximation from (.16)..1 Signals and charge spread in detectors with resistive elements In this subsection we calculate the signals induced on a readout pad or a readout strip in presence of a resistive layer, either as a bulk resistive layer touching the readout structure figure 9 or as a thin resistive layer that is insulated from readout pads(figure ). Like we discussed before the time dependent weighting fields for a pad of (a) Weighting field for a geometry with a resistive layer having a bulk resistivity of ρ = 1 σ. (b) Weighting field for a geometry with a thin resistive layer of value R. dimension w x and w y centred at zero and a infinitely long strip of width w x and w y centred at zero, can be written : E z w(x, y, z, t) = V w g for both geometries. 4 π E z w(x, y, z, t) = V w g ( ) ( ) ( ) ( ) x w x y w y h(k, z, t) cos k x sin k x cos k y sin k y dk x dk y (.17) g g g g k x k y π ( ) ( ) x w x h(k, z, t) cos k x sin k x dk x dk y (.18) g g k x k y.1.1 Layer with bulk resistivity For a layer with bulk resistivity of ρ = 1/σ the expression of h(k,z) (for <z<g) is : ( ( h(k, z, t) = kcosh k 1 z )) ( ɛr δ(t) g D(k) + 1 ) b 1 (k)e tf 1 (k) τ τ (.19) ( D(k) = sinh k b ) ( cosh(k) + ɛ r cosh k b ) sinh(k) (.13) g g

50 . Electric fields,weighting fields and signals in detectors including resistive materials 36 b 1 (k) = ( ) sinh k b g cosh(k) D(k) f 1 (k) = ( ) sinh(k) cosh k b g D(k) (.131) with τ = ɛ /σ = ɛ ρ We investigate the geometry where the ground plate at z = b is segmented into infinitely long strips of width w x. We assume a pair of charges Q,-Q produced at t = at z =, the charge Q does not move and the charge -Q moves from z = to z = g with uniform velocity z(t) = vt = gt/t, <t<t,t = g/v(figure ). The current is : I(t) = Q E w (x, z(t ), t t )z (t )dt = Q E w (x, z(t ), t t )g/t dt t < T (.13) V w V w I(t) = Q V w T E w (x, z(t ), t t )z (t )dt = Q T E w (x, z(t ), t t )g/t dt t > T (.133) V w And so : I(t < T ) = Q T ( ) ( ) π cos x w x k x sin k x g g [ ɛ r cosh (k kt/t ) + D(k) e tf1 τ (f 1 cosh(k) + τ T +b ksinh(k)) f 1 cosh ( ) k k t T K τ T sinh ( ) k k t T 1 dk k τ (.134) T f 1 I(t > T ) = Q T ( ) ( ) π cos x w x k x sin k x g g b 1 e t T τ f1 e T f1 τ (f 1 cosh(k) + k τ k τ T f 1 T sinh(k)) f 1 dk (.135) For a very high resistivity limit τ the layer represents and insulator and : lim τ I(t < T ) = Q T ( ) ( ) π cos x w x ɛr cosh ( ) k k t T k x sin k x dk g g D(k) lim I(t > T ) = (.136) τ For the case where the layer represents a perfect conductor the expression becomes : lim I(t < T ) = Q ( ) ( ) ( ) τ T π cos x w x cosh k k t k x sin k x ( T ) dk g g sinh(k)cosh k b g This last expression is correct if the strips are truly grounded. lim I(t > T ) = (.137) τ For any realistic setup where the strips are connected to readout electronics and therefore have a finite resistance to ground, the signal will spread to all the strips together with the bulk behave as one single node. The result is therefore correct only to levels of conductivity σ where the impedance between the strips is significantly larger than the input resistance of the amplifier. Figures 9,3,31 show the induced current signals given above on a central strips w x = 4g and the first neighbouring strip centred at x = 4g for different values of conductivity, for a different time constants τ. The figures show in dashed lines also the limiting cases for a very large and very small values of τ.

51 37. Electric fields,weighting fields and signals in detectors including resistive materials (a) x = (b) x = 4g Figure 9: Uniform charge movement from for z = to z = g, with ɛ r = 1,w x = 4g,b = g,t = 1T. (a) x = (b) x = 4g Figure 3: Uniform charge movement from for z = to z = g, with ɛ r = 1,w x = 4g,b = g,t = T. (a) x = (b) x = 4g Figure 31: Uniform charge movement from for z = to z = g, with ɛ r = 1,w x = 4g,b = g,t =.1T. First we observe that all the signals are unipolar, which is due to the fact that the charge that is flowing in the resistive bulk layer in order to compensate the charge Q sitting in the surface of the resistive plate. is truly coming out of the readout strips. For τ = T the signal is significantly affected and develops a long tail for t>t due to the flow of charge compensating the point charge on the surface. The smaller the conductivity, the longer(smaller) is the tail of the signal, for τ = 1T. For short time constants of the resistive layer the signal on the central strip is large and has short tail and the crosstalk to the neighbor strips increases(for τ =.1T ). Next we will give the result for a pair of charges created at z = g and the charge -Q moving from z = g to z = with z(t) = g gt T.

52 . Electric fields,weighting fields and signals in detectors including resistive materials 38 I(t < T ) = Q T ( ) ( ) ( π cos x w x ɛ r cosh(kt/t k x sin k x g g D(k)) e tf1 τ f 1 cosh ( ) k t T + k τ + b 1 f T sinh ( k t T 1 k τ T f 1 )) dk (.138) I(t > T ) = Q T π cos(k x x g ) sin ( ) w x k x g b1e t T τ f1 e T f1 τ + k τ f 1 T sinh(k) f 1 cosh(k) k τ T f 1 dk (.139) For a very high resistivity limit τ the layer represents and insulator and : lim τ I(t < T ) = Q T ( ) ( ) π cos x w x ɛr cosh ( ) k t T k x sin k x dk g g D(k) lim I(t > T ) = (.14) τ For the case where the layer represents a perfect conductor the expression becomes : lim I(t < T ) = Q ( ) ( ) τ T π cos x w x cosh ( ) k t T k x sin k x ( ) dk g g sinh(k)cosh k b g lim I(t > T ) = (.141) τ.11 Layer with surface resistivity Last we discuss the example for a thin layer of surface resistivity R on top of an insulating layer. The expression of h(k,z) : ( h(k, z, t) = k cosh k(1 z ) ( g ) ɛr δ(t) D(k) 1 ) b (k)e tf (k) T (.14) T D(k) = sinh(k ( ) ( b ) cosh(k) + ɛ r cosh k b ) sinh(k) (.143) g g ( ) ɛ r sinh k b g sinh(k) b (k) = k D (k) ( ) sinh(k) sinh k b g f (k) = k D(k) (.144) with T = ɛ Rg is the time constant associated with the resistive layer in the given geometry. For a pair of charges Q,-Q produced at t = at z =, the charge Q does not move and the charge -Q moves from z = to z = g during a time T, we find : I(t < T ) = Q T ( ) ( ) π cos x w x k x sin k x g g [ ɛ r cosh (k kt/t ) + D(k) e tf τ (f cosh(k) + τ T b ksinh(k)) f cosh ( ) k k t T K τ T sinh ( ) k k t T dk k τ (.145) T f I(t > T ) = Q T ( ) ( ) π cos x w x k x sin k x g g b e t T τ f e T f τ (f cosh(k) + k τ k τ T f T sinh(k)) f dk (.146)

53 39. Electric fields,weighting fields and signals in detectors including resistive materials The limited case for high resistivity is equal to the previous subsection s where there is only an insulating layer. For the limited case for small resistance R, the I(t) becomes zero, since the resistive layer turns into a metal plate that shields the strips from the charges Q and Q. The signals for a central strip of width w x = 4g as well as the neighbouring strips at x = 4g and x = 8g as shown in figures 3-36 for different values of R and different time constant T. (a) x = (b) x = 4g (c) x = 8g Figure 3: Uniform charge movement from for z = to z = g, with ɛ r = 1,w x = 4g,b = g,t = 1T. (a) x = (b) x = 4g (c) x = 8g Figure 33: Uniform charge movement from for z = to z = g, with ɛ r = 1,w x = 4g,b = g,t = T. (a) x = (b) x = 4g (c) x = 8g Figure 34: Uniform charge movement from for z = to z = g, with ɛ r = 1,w x = 4g,b = g,t =.1T.

54 . Electric fields,weighting fields and signals in detectors including resistive materials 4 (a) x = (b) x = 4g (c) x = 8g Figure 35: Uniform charge movement from for z = to z = g, with ɛ r = 1,w x = 4g,b = g,t =.1T. (a) x = (b) x = 4g (c) x = 8g Figure 36: Uniform charge movement from for z = to z = g, with ɛ r = 1,w x = 4g,b = g,t =.1T. In case the time constant T is large, the effect of resistivity disappears and the case of T = 1T shows signal shapes very close to the on from the previous section for large values of τ. For decreasing resistivity and so T, we see that the signal on the central strip starts to be "differentiated" and develops an undershoot and the crosstalk to the other strips increases. The signal are strictly bipolar. This is due to the fact that the current compensating the point charge Q is entirely flowing inside the thin resistive layer and no net charge is taken from or is arriving at the strips. Next we will give the result for a pair of charges created at z = g and the charge -Q moving from z = g to z = I(t < T ) = Q T ( ) ( ) π cos x w x ɛr cosh(kt/t ) e tf T f cosh(k t T k x sin k x b f ) + k T T sinh(k t T ) ]dk g g D(k) k τ (.147) T f I(t > T ) = Q T ( ) ( ) π cos x w x k x sin k x g g be t T T f e T f T + k T T f sinh(k) f cosh(k) dk k T (.148) T f

55 41 3. Micromegas Simulation with Ansys Maxwell 3 Micromegas Simulation with Ansys Maxwell In this Section and the next one we will try to analyze some structures of Micromegas Modules. Specifically in this Section we will create some variety of MM, with the help of Ansys Maxwell 14.. Our aim is to check how the capacitance between the strips are changing for every one of those modules. Our analysis will focus mostly for the values between the central strip and the two neighbor in all those cases. Maxwell is a high-performance interactive software package that uses finite element analysis(fea) to solve twodimensional and three-dimensional (3D) electric,magnetostatic, eddy current and transient problems. In this section in order to create the modules we need, we solve two-dimensional problems. 3.1 Structure of the creating Modules by Maxwell In the following figure that was taken by a microscope we can see a form of the Micromegas Module that we examine in our work. Figure 37: Micromegas Module by microscope The Micromegas modules that we are creating in this section are consist of the following parts : (A) the bottom is the element Fr4 with dielectric constant 4.4. (B) the readout strips that they are important to read the signal and they are creating from copper. (C) the area between the Resistive and the Readout strips consisting of insulating material PC15 Dupont with dielectric constant 3.5. (D) the Resistive strips that they consisting of Resistive material with conductance.59 siemens/m and dielectric constant 1. (E) a support area of the Module(we will not focus on this). In the next subsection we will give more details about those parts and their dimensions. 3. Micromegas Modules In the following figure is shown the general structure of the Micromegas module that was built by the Ansys Maxwell program. The following module has 3 Resistive strips and 3 read out strips.

56 3. Micromegas Simulation with Ansys Maxwell 4 Figure 38: Micromegas Module with 3 Resistive and 3 readout strips. As we discussed in the previous subsection all of our models are included by the fr4, the Dupont and a number of Resistive and Readout Strips. We will also add a Mesh(line) from Iron on a distance 18 µm from the Resistive strips and also we put the Ground on the down part of the Fr4. As for the size of its components Ansys Maxwell give us a great variety of options to change the dimensions on the 3 axis and "play" with them. We will hold the -D option for our structure. So to have a basis for all of the Modules we are going to discuss, we will give the the following dimensions to the elements : The resistive strips have width 3 µm and thickness 15 µm The Read Out strips have width3 µm and thickness 17 µm The Fr4 has 5 µm thickness. The kapton has 85 µm thickness. The line has 7 µm thickness. Between the kapton and the line we left a space of 18 µm. This section is concerned with 5 strips and 9 strips Modules. Our Modules were slipped on two categories the LM Modules and the SM Modules. The different between those two Modules is the pitch. The LM Module has 45 µm pitch and the SM Module has 45 µm pitch. Figure 39: LM Module

57 43 3. Micromegas Simulation with Ansys Maxwell Figure 4: SM Module For this section we built with Maxwell 7 Modules : two LM Modules with 5 and 9 strips each two SM Modules with 5 and 9 strips each one 5 strips Module with 1 µm distance between the strips and finally two Modules with 5 strips each for those two categories, but this time we added as new material glue. By doing all these Modules we want to see how the capacitance between the strips change, by add more materials on the Module or increase the distance of some of the components Micromegas 5 strips LM Module The first module is discussed is a 5 strips(resistive and Readout) LM Module. In the end of the analysis with Maxwell, the program will give us a panel with the capacitance of all the strips.we focus on the central strips that in this case is Readout Strip 3 and the Resistive strip 3(Res-Ro) and the capacitance between this strip and Readout Strip (Ro-Ro), Resistive Strip and the capacitance between Resistive strip 3 and (Res-Res). The values we got from this module are : Res-Res :.6861 Res-Ro : Ro-Ro : 3.39 Figure 41: 5 strips LM Module In the following table Maxwell gathered all the capacitance between the strips. The abbreviations Res and Ro are for Resistive and Readout strips. As for the the numbers of the strips were selected for the row that they were placed on the Module.

58 3. Micromegas Simulation with Ansys Maxwell 44 Figure 4: Table of capacitance for 5 strips LM Module. 3.. Micromegas 5 strips SM Module The process in this module is the same with the difference that in the SM Module the distance between the strips is less than before. Res-Res : Res-Ro : Ro-Ro : 36.7 Figure 43: 5 strips SM Module Figure 44: Table of capacitance for 5 strips SM Module. If we compare the 5 LM Module with this we can observe a different in the 3 values. The Res-Res and the Ro-Ro values are higher in the SM Module, but the Res-Ro value are lower Micromegas 9 strips LM Module In this case we increase the number of strips from 5 to 9 in the strips and their distances from each other are the same with the 5 strips LM module. Res-Res : Res-Ro : Ro-Ro : 36.7

59 45 3. Micromegas Simulation with Ansys Maxwell Figure 45: 9 strips LM Module Figure 46: Table of capacitance for 9 strips LM Module Micromegas 9 strips SM Module This module also is the same with the 5 strip SM Module but with 9 strips instead of 5. Res-Res :.6861 Res-Ro : Ro-Ro : 3.39 Figure 47: 9 strips SM Module Figure 48: Table of capacitance for 9 strips SM Module. 3.3 More cases of micromegas Modules As the next step we examine some more cases of Micromegas Modules with different distances between the strips and what happens if we put more elements in one of these Modules Micromegas 5 strips LM Module with 1 µm We will start with reducing the distance between the Resistive strips and the Readout Strips at 1 µm. In the next table we have the results for the central Strip. Res-Res : 6.9

60 3. Micromegas Simulation with Ansys Maxwell 46 Res-Ro : Ro-Ro : 45.9 Figure 49: 5 strips 1 µm LM Module Figure 5: Table of capacitance for 5 strips 1 µm LM Module Micromegas 5 strips LM Module with Glue In this last two cases we have the same 5 strips LM and SM modules but with an extra glue layer above the Dupont. The layer is actually Glue from teflon with Relative Permittivity.8. Res-Res : 4.8 Res-Ro : Ro-Ro : 8.4 Figure 51: 5 strips LM Module with glue Figure 5: Table of capacitance for 5 strips SM Module with glue.

61 47 3. Micromegas Simulation with Ansys Maxwell Micromegas 5 strips SM Module with Glue In this last case we have the same 5 strips LM module but with an extra glue layer(thickness of 35 µm above the Dupont). Res-Res : 6.7 Res-Ro : Ro-Ro : Figure 53: 5 strips SM Module with glue Figure 54: Table of capacitance for 5 strips SM Module with glue. 3.4 Coclusion In the following table are included the values for the central strips for all the previous cases. Table 3: Capacitances for different modules of Micromegas 5 LM 5 Sm 9 LM 9 SM 5 LM with Glue 5 SM with Glue 1 µm Res Res Ro-Ro Res- Ro From the table some conclusions have been drawn : For the 5 strips Modules(with Glue or without) the capacitance between the Resistive strips and the Readout Strips of 1 µm Module has higher values from the SM Module and this has higher values than LM Module but the values between the Resistive strips and the Readout strips are upside down. We can say that as long we reduce the distance between the Resistive and the Readout strips the values of the Capacitance increase and also the values between the Resistive and the Readout strips reduce. For 9 strips Module we have the same results with the 5 strips Modules and the same values. It seems that the most important thing is the central strips and so the results did not change. If we compare the 5 LM and SM Modules(with and without Glue) we can make a conclusion that we every layer we add the value between the Resistive and between the Readout strips starting to reduce.

62 4. Micromegas Simulation with LTspice 48 4 Micromegas Simulation with LTspice 4.1 Introduction In this Section we will continue the analysis with the help of LTspice. With LTspice we will create the circuits that correspond to the Maxwell modules of the previous Section. We will also observe the signal from the central strips and those around it and how it changes for different values of capacitance. As a basis We use the circuit by Ludwig-Maximilians-Universitat Munchen - Lehrstuhl Schaile(figure 55). Figure 55: Ludwig-Maximilians-Universitat Munchen - Lehrstuhl Schaile Module. As it clear from the figure we applied a current pulse. Table 4: variables of current I Current Values(Amper) I 1 A I 1 µα T dellay 1 na T rise.1 na T fall.1 na T on 18.4 na T period.1 na The capacitors C9 and C3 correspond to the capacitances between the resistive strips. C11 and C4 correspond to the capacitances between the readout strips. C1,C11,C1 are the capacitances between resistive and readout strips. The capacities C5,C7,C6,C8,C1,C13 correspond to the readout for our circuit and we use the values from our prototype model(1 nf and, 4 pf). 4. Spice Simulation with LM and SM Modules As in the previous section we working on 5 and 9 strips modules LM and SM. And for every one of them we draw the figures for the output signal of the central and neighbor strips,output voltage and the output current. For those figures we will obtain the percentage of diffusion from the central to the neighbor strips. In all those figures the line with the bigger depth represents the central strip and the others two are for the neighbor strips. 5 strips LM Module First we start with the 5 strips LM Module(Figure 56).

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