Theory of Chambers: Plane Geometries

General description of the researches

We developed theory to compute the fields generated within Resistive Plate Chambers (RPC's) and  the signals induced in the electrodes of RPC's by a point charge moving in  the gas gap with constant velocity along a trajectory perpendicular to the electrode planes.  The induced currents can be determined with the help of Ramo's theorem (1939) *). This theorem expresses the current   induced by a moving charge by the product of the charge motion with a static field called weighting fields, which would be generatred in the same configuration in the absence of the charge by a static Voltage applied to the electrode.  Ramo's proof assumed isolating dielectrics between the electrodes. RPC's comprise several layers, whose resistivity is high but not infinite.  Indeed, it is so high that  the resulting decay rate is very long as compared to the times, which the particles or their fields need to cross the structure. Riegler (2002) of CERN generalized Ramo's theorem to the case where the dielectric has a low conductivity; for this, he used a quasi-static theory for weakly conducting media presented by Heubrandtner and Schnizer (2002).

In another approach one may compute the field generated in the configuration by the moving charge; from this time-dependent field one may also derive the currents induced in the electrodes. This  approach  was started by Schoepf and Schnizer in 1992.

The investigations just described show that the conductivity and permittivity of the layers in-between the gas gap and the electrodes is quite small for the values of these quantities used in real RPC's. So it is mainly the electrode structure which determines the weighting field which may be used to calculate the signal.  We derived expressions for the potential and field present in simple two-dimensional models for such structures by conformal maps. These have implemented in user-friendly Mathematica programs calculating and plotting potential and field distributions and signal currents in the electrodes.

Standard detector physics simulations can only be performed by numerical methods. Such studies involve also the space charge dueto  the electron cloud (avalanche) present and growing in the "on" time of the detector.  For dynamic calculations of the electric field of the space charge, it is very useful to have analytic expressions (Green's functions represented as series and/or integrals) for the field of a point charge in such a layered structure. These expressions have been rendered more useful by improving their convergence properties.

*)  W. Blum, L. Rolandi, Particle detection with drift chambers. Springer, Berlin, 1994.

Ramo's theorem and the quasi-static approximation

The generalization of Ramo's theorem for weakly conducting media and applications to signals induced in infinite or strip electrodes were derived in:

             W. Riegler,
             Induced signals in resistive plate chambers.
             Nucl. Instrum. Methods Phys. Res., A : 491 (2002) pp.258-271

The method used in this paper is described in the following papers:

             Th. Heubrandtner, B Schnizer,
             The quasi-static electromagnetic approximation for weakly conducting media
              Nucl.Inst.Meth. Phys. Res. A478 (2002)  444-447

            Th. Heubrandtner, B. Schnizer, W. Riegler,
            The quasi-static approximation for weakly conducting media and applications.
             Proceedings, 11th Internat. IGTE Symp. on Num. Field Calc. in Electr. Eng.,
             Seggau Castle, Sept. 13 - 15, 2004 , pp. 138-143.
             pap83

             Th. Heubrandtner, B. Schnizer, L. Dedek,
             A quasi-static method for solving  transient problems in weakly conducting media.
             Kleinheubacher Berichte 43 (2000) 445-451
             pap76


Time-dependent fields due to point charges moving in chamber structures


Such fields were investigated in the following papers:

            Th. Heubrandtner,
             Theoretical Models for Signal Generation in Resistive Plate Chambers.
             Doctoral dissertation, Faculty of Science, Technical University of Graz, May 1999.
             HeubraDiss  in papers

              H. Schöpf,  B. Schnizer
             Theory describing cathode signals from charges moving in counters with a poorly conducting
              cathode.
              Proc. 1992 Wire chamber Conf. Vienna, 18-21 February 1992.
              Nucl.Inst.Meth. Phys. Res. A323 (1992), 338-344

             Th. Heubrandtner, B. Schnizer, H. Schöpf,
             Signals in a Resistive Plate Chamber
             Kleinheubacher Berichte 41 (1998) 484 - 489

Conformal maps and Mathematica programs for computing wheighting fields

The following two-dimensional models were investigated by conformal maps and the resulting formulae implemented in Mathematica programs stored in the subdirectory Mathematica programs for weighting fields  :

  1. An infinite empty plane condensor, whose upper electrode contains a strip with an impressed Voltage.The remaining parts or electrodes are grounded.  The corresponding Mathematica notebooks  are:
    CondVoStr.nb and CondShVoStr.nb
    There is a LongWriteUp notebook in which the conformal map is also discussed: CondVoStrLW.nb

  2. An infinite empty plane condensor; the upper electrode is semiinfinite and a Voltage is supplied to it;
    the lower electrode is grounded. The corresponding notebook is:
    ShSemInfCond.nb

  3. An infinite empty plane condensor, whose upper electrode is split in two by a gap of given finite width. The semiinfinite electrode at the right and the lower one are grounded. A Voltage is supplied to the other semiinfinite electrode.   The corresponding notebook is:
    SplitCond.nb
    In the same subdirectory there is a LongWriteUp notebook in which the conformal map is also discussed: SplitCondLW.nb

  4. The program subdirectory contains several notebooks in which the different models are compared:
    Comparison.nb,  RS-Comp-SemInf-Strip.nb,  RS-Comp-Split-SemInf-Strip.nb, RS-Comp-Split-SemInf.nb, RS-Comp-Split-Strip.nb.  

    This comparison is dynamic inasmuch as the widths of the gap and if the strip are made to vary in the notebook Animation.nb or in the GIF-file Animation.gif .

The theory and descriptions of the programs are given in the following papers:

            Th. Heubrandtner, B. Schnizer, G. Schweitzer,
            Simple Models for RPC Weighting Fields and Potentials.
            Nucl. Instrum. Methods Phys. Res., A : 535 (2004) no.1-2, pp.454-457

            St.Rossegger, B. Schnizer, G. Schweitzer,
            Comparison of simple models for RPC weighting fields and potentials.
            Nucl.Inst.Meth. Phys. Res. A 535 (2004)  554 - 557
            Revised and extended unpublished version of previous paper
            pap82a  in papers

Mathematica programs ploting potentials and fields in similar and other configurations  can be found at the
page Conformal Maps

Green's functions for layered structures and space charge studies

           
            Th. Heubrandtner, B. Schnizer, C. Lippmann, W. Riegler,
            Static electric fields in an infinite plane condensor with one or three homogeneous layers.
            Nucl. Instrum. Methods Phys. Res., A : 489 (2002) no.1-3, pp.439-443
            (= CERN-EP-2002-004 ; Geneva : CERN , 8 Jan 2002)

            Th. Heubrandtner, B. Schnizer, C. Lippmann, W. Riegler,
            Static electric fields in an infinite plane condensor with one or three homogeneous 
            layers.
            Report CERN-OPEN-2001-074, 31 Oct. 2001
            (This is an extended preprint version of the previous paper)

            C. Lippmann, W. Riegler, B. Schnizer
            Space charge effects  and induced signals in resistive plate chambers.
            Nucl. Instrum. Methods Phys. Res., A : 508 (2003), pp.19 - 22

  1. papers
  2. Mathematica programs for weighting fields