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A study of breakdown in microstrip gas chambers
Nuclear Instruments and Methods in Physics Research A 348 (1994) 356-360 North-HoUand
NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH SechonA
A study of breakdown in microstrip gas chambers I. Duerdoth *, S. Snow, R. Thompson, N. Lumb Department of Physics and Astronomy, Schuster Laboratory of The Manchester University, Manchester M13 9PL, UK
A problem that occurs with MicroStrip Gas Chambers (MSGCs) is breakdown between the electrodes. We have used two techniques to simulate and study this. The three dimensional electric field has been modelled by a computer simulation in which a finite number of point charges are moved to their equilibrium positions within the electrodes. Secondly, a real 40 x scale model of our MSGC was operated at 1/40th of normal pressure. Results from both techniques lead us to a new shape for the ends of the cathode strips.
1. Introduction The MSGC is today a developing technology which is expected to replace the wire chamber as an economic, precise position sensitive detector for tracking at high rates [1]. The principle [2] is now well known; alternating broad cathode and narrow anode strips are etched in a thin ( ~ 1 Ixm) metallised layer on a substrate. The narrow anodes act like wires in a conventional chamber, providing gas amplification for electrons drifting in from the gas volume. The strong dipole field between anode and cathode quickly clears the positive ions from the anode region, producing a fast signal. One problem suffered by MSGCs is that the highest gain that can be reached before breakdown of the gas in the anode-cathode gap is about 103 to 104. At these gains it is hard to detect the signal from a single primary ionisation in the gas and this may limit the efficiency of the thin gas-gap MSGCs which are needed for high rate applications. When a MSGC is run near its gain limit there are occasional sparks which can erode the electrodes (Fig. 1). This figure immediately shows that the breakdown is localised at the ends of the cathodes, which is also the region where the electric field near the cathode surface is at its maximum. This maximum gain is influenced by the gas mixture used [3], the substrate type (thin layer or bulk conductivity) [4] and the thickness [5] and shape of the electrodes. We have chosen to concentrate on optimising the shape of the cathode ends because this does not add to the cost and does not influence the other performance characteristics of the detector. We have used a 3-D simulation of the electrostatic field and a 40 X life-size scale model to understand the breakdown mechanism and find an improved shape for the cathode ends. The reference design, which we hope to improve on, is
an MSGC with 100 p.m wide cathodes and 10 p~m wide anodes at a pitch of 300 p.m made from 1 p.m thick aluminium. The cathodes have semi-circular ends. The substrate has some bulk conductivity sufficient to prevent charging up and we believe that we can ignore the different dielectric permittivity of gas and substrate because we operate with a field configuration in which few field lines cross the gas-substrate surface. The gas is a r g o n / i s o b u t a n e 7 5 / 2 5 and we know that a real chamber of this design has a gain of about 3000 at 700 V and it breaks down at about 720 V.
2. Electrostatic simulation We have used a multi-particle approach based on the ideas of ref. [6] and extended it to describe many conductors at different voltages. First we describe our algorithm in some detail while the next section covers the use and interpretation of our final electric field maps.
* Corresponding author. 0168-9002/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0 1 6 8 - 9 0 0 2 ( 9 4 ) 0 0 5 3 0 - K
Fig. 1. Damage caused by sparks to the electrodes of a MSGC.
L Duerdoth et al./Nucl. Instr. and Meth. in Phys. Res. A 348 (1994) 356-360
357
2.1. Method This simulation only describes a system of perfect conductors surrounded by a uniform dielectric. We will use the indices k and l to label conductors and i and j to label particles. Suppose that we wish the conductors to be at potentials Vk. Each conductor is populated with a number of identical point particles of charge Q~. For a given distribution of particles the following quantities are calculated: is the vector from particle i to particle j. eij = Ei ÷ j Q i / I Rij I is the potential energy of a unit charge at the position of particle j and Ey = QjPy is the potential energy of the particle itself. Fj = ~]i ÷ j Q i a j R i j / [ Riy [ 3 is the force on the particle j. Vffv is the average value of Py of the charges in conductor k. VRMs is the root mean square deviation of the Pjs from the desired voltage of their conductor, Vk. We normalise V~ Ms to the scale of the largest voltage difference in the system, in this case 700 V. With this information availible, one of three possible steps can be used to converge towards the desired voltages (i.e. to minimise vRMS). 1) Move each particle some distance in the direction of the force Fj. The distance is chosen to be some small fraction f times the distance of the particle from its nearest neighbour. When a particle comes up against the surface of the conductor it is moved in the direction of the component of F which is perpendicular to the surface. The fraction f is initially 0.1 and is reduced as the system nears equilibrium. 2) Move just one particle in each conductor. Choose the particle with the highest value of Ej in each conductor and place it near to the one with the lowest Ej in the same conductor. 3) Change the charges of all of the particles such that the mean potential in each conductor VAv is at the desired value Vk. This can be done by computing the capacitance matrix Ckt = ~i~.j~:i 1 / I Rij l, where index i runs over all charges in conductor k and index j runs over all charges in conductor 1. The inverse of Ckt multiplied into V~ gives the new values of the charges for each conductor. The initial positions of the particles are randomly distributed within the volume of their associated conductors and ~ 1000 iterations of the above steps are used to reach a solution. Most of the work is done with iterations of type 1' and 3 but type 2 is useful at the early stages for removing long-range variations of potential which take a long time to smooth out with type 1 steps. Steps of type 1 and 3 will only converge towards the desired solution if the diagonal terms of the capacitance matrix are large compared to the off-diagonals. This condition can be achieved by increasing the number of charges on the conductors.
f
•
I
.t
q
•
I
•
I
•
I
+ o
I,
Fig. 2. The final distribution of charges in our simulation of a MSGC of conventional geometry. The points give the positions of the charges and the thin lines are the boundaries of the conductors.
Another trick that we have used is to start out with a relatively small number of particles ( ~ 50) in each conductor and when these have reached equilibrium to double their number by placing a new particle in the space between each particle and its neighbour. This way the new particles start out from a position which is close to equilibrium over long ranges and only local rearrangements are needed to get back to a good solution. In addition, one physical conductor may be split up into a number of smaller parts for the purposes of the simulation and the parts which we are interested in may be simulated in greater detail than the rest. The quality of the electric field solution is indicated by V RMs. The results of the next section are derived from a distribution of 5800 particles in the region of interest and 700 particles on the surrounding conductors which is shown in Fig. 2. This simulation used around 30 hours cpu time on a HP735 (40 Mflops). The v R M S s of the 14 conductors are in the range 1-3%. While searching for an improved shape of cathode ends we have used a similar amount of computer time to produce field strength maps for a more complicated arrangement of conductors such as Fig. 5, in which the V RMs values of the important conductors were in the range 3-7%. II. MICROSTRIP GAS CHAMBERS
358
L Duerdoth et al. /Nucl. Instr. and Meth. in Phys. Res. A 348 (1994) 356-360
2.2. Results
Fig. 3 shows a contour map of the electric field strength near the end of a cathode strip of the standard design. We are looking in the plane parallel to the conductors but 5 Ixm inside the gas gap. The contour interval is 1 V p,m-1 • The highest field near the tip of the cathode is 16.8 V Ixm- 1 while the field near the straight side of the cathode is 10.7 V txm -1. We believe that it is these high fields which are causing the breakdown because an electron liberated from the surface of the cathode will start a significant avalanche within 10 Ixm and photon feedback cannot be absorbed within this short distance and will liberate another electron. This is investigated by looking at examples of field lines originating at the tip and edge of the cathode and indicated by plain and dashed lines on Fig. 3 respectively. We chose field lines which leave the cathode at an angle of about 45 ° to the plane of the conductors and loop out into the gas volume before returning to the anode. Fig. 4a shows the field strength versus the distance along the field lines while Fig. 4b shows the value of the first Townsend coefficient, a , corresponding to these fields• The Townsend coefficients were computed with the program M A G B O L T Z [7] and are only approximate for
/ \
/ Fig. 3. Contours of equal electric field strength in the plane inside the gas of an MSGC 5 p~m above the electrodes. The contour spacing is 1 V p,m-1. The plain and the dashed lines are field lines from the tip and edge of the cathode respectively.
o ._o 3O
l
(a)
i
I
E ~ 20
~-o 10 ~z 0
i
J 0
20
40
60
80
1O0
t 20
Distance (microns) ~ 1.5 L E ~ t "~ .c ~" 0,5 ~,_ 0 c ~o J-
o
15
(b)
20
:_
(c)
0
20
i
40
60
80
1O0 120 Distance (microns)
_J
,-° o. -6 10 "_o ~ 5 -¢
0
, , , I , , ,
40
60
80
100
120
Distance (microns)
Fig. 4. (a) The field strength plotted as a function of the distance along the two example field lines from the cathode end (plain) and the cathode edge (dashed). (b) The corresponding Townsend first coefficient. (c) The integral of the first Townsend coefficient. fields greater than 10 V p,m -1 because good data for isobutane is not available. However, we have found that the relative gains in different parts of the chamber have little sensitivity to the shape of the a ( E ) curve within the range of variation that we have studied. Fig. 4c shows the integral of the Townsend coefficient along the field lines. One can see from Fig. 4c that an electron drifting in from the low field region to a typical part of the anode (right hand half of the dashed line) will be amplified by a factor e 9. This agrees reasonably well with the gain of e 8 = 3000 measured with the real detector and gives us confidence that the other gains predicted by this simulation will be correct within an order of magnitude. For example we predict that in the first 10 p~m of travel away from the cathode surface an electron will produce an avalanche of e 72 = 1300 electrons if it came from the tip of the cathode or e 43 = 74 electrons if it came from the straight edge of the cathode.
3. I m p r o v e d d e s i g n
Our aim is to increase the gain that an MSGC can reach before breakdown and it is clear that one way to do this is
L Duerdoth et al./NucL Instr. and Meth. in Phys. Res. A 348 (1994) 356-360
359
4. Scale model
( ! • i • ! • it * •
, •
°
, o o ° *
° ° i
•
• o *
i i
Fig. 5. An example of charge distribution on the improved cathode design which slightly lowers the maximum electric field near the cathode surface.
to reduce the maximum field that occurs near the cathode surface. This could be done by thickening and rounding off the edges of the cathodes, which would require a change to our etching method, or alternatively by changing the mask pattern. We have changed the pattern by terminating the cathodes with " b l o b s " which have larger diameter than the cathode width in the working part of the detector (Fig. 5). In order to leave space for the anodes, alternate cathodes must be given a neck and extended beyond the other cathodes. We have found that the amount of improvement possible from this scheme is so small that it cannot be comfortably measured when the accuracy of our simulation is only + 3%. The simulation did tell us though that the field at the tip of the short cathodes can be reduced by increasing the size of blobs on the long cathodes, because the repulsion of these two areas of negative charge tends to prevent charge from accumulating at the tip of the short cathode. The difficulty of making a sufficiently accurate simulation persuaded us to go to the scale model described in the next section.
The properties of a gas relating to its breakdown under an applied electric field are always functions of the ratio of field strength to gas pressure. This means that we can scale the dimensions of our MSGC up by some factor and the gas pressure down by the same factor and expect that its behavior will be unchanged. The one thing that is changed by this scaling is the time duration of the processes. If there was an important fixed time scale involved in breakdown, for example the lifetime of a metastable state, then our scale model would not reproduce the real chamber. We have used a scale factor of 40 which enables us to build test structures easily on printed circuit board and test them in a low pressure chamber of reasonable dimensions (20 × 20 cm) with transparent windows. The breakdown voltage was measured by turning up the applied voltage slowly and recording the point at which a visible discharge appeared. This was repeated several times to obtain the error estimate. The scale model of our standard design breaks down at 915 + 10 V at the cathode ends whereas the real chamber breaks down in the same place at about 720 V, showing that the scaling law works moderately well. Many other test structures have been tried, all following the theme of "alternating blobs" which was described above. The overall picture is that it is hard to find a structure which is any better than the plain semi-circular ends for terminating the cathodes. However, we have found one pattern (Fig. 5) which is slightly better, holding up to 940___ 10 V before breakdown. The " r u l e s " of a better pattern seem to be that the anode-cathode gap must always be larger than the one in the sensitive part of the detector, the cathode should not be necked down below about 50% of its width and the blobs on the longer cathodes should be as large as possible, even though it is not here that the breakdown occurs.
5. Conclusion A multi-particle method of simulating the 3-D electrostatic field around a fairly complex group of conductors at chosen voltages is described. This simulation can produce a field map accurate to about 2% in the space between the conductors, provided that one does not look too close to the conductors where the influence of individual point charges becomes noticeable. Combining our field map with various parametrisations of the first Townsend coefficient has shown that when the amplification at the anodes is at a useful value ( ~ 10 000) then there is inevitably an amplification of order 1000 within 10 p~m felt by an electron leaving the cathode tip. A small ( ~ 3%) increase in the maximum sustainable voltage can be achieved by changing the shape of the cathode II. MICROSTRIP GAS CHAMBERS
360
L Duerdoth et aL /Nucl. Instr. and Meth. in Phys. Res. A 348 (1994) 356-360
ends. This leads to a worthwhile increase of about 50% in the m a x i m u m gain o f the chamber.
References [1] RD28 Status Report, CERN/DRDC/93-34. [2] A. Oed, Nucl. Instr. and Meth. A 263 (1988) 351.
[3] M. Geijberts et al., Nucl. Instr. and Meth. A 313 (1992) 377. [4] F. Angelini et al., INFN P I / A E 93/07 (1993), submitted to Nucl. Instr. and Meth. [5] Yu. Pestov and L. Shekhtman, BINP 93-59 (1993). [6] H. Van der Graaf and J.P. Wagenaar, Nucl. Instr. and Meth. 217 (1983) 330 and A 252 (1986) 311. [7] S. Biagi, Nucl. Instr. and Meth. A 283 (1989) 716.
NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH SechonA
A study of breakdown in microstrip gas chambers I. Duerdoth *, S. Snow, R. Thompson, N. Lumb Department of Physics and Astronomy, Schuster Laboratory of The Manchester University, Manchester M13 9PL, UK
A problem that occurs with MicroStrip Gas Chambers (MSGCs) is breakdown between the electrodes. We have used two techniques to simulate and study this. The three dimensional electric field has been modelled by a computer simulation in which a finite number of point charges are moved to their equilibrium positions within the electrodes. Secondly, a real 40 x scale model of our MSGC was operated at 1/40th of normal pressure. Results from both techniques lead us to a new shape for the ends of the cathode strips.
1. Introduction The MSGC is today a developing technology which is expected to replace the wire chamber as an economic, precise position sensitive detector for tracking at high rates [1]. The principle [2] is now well known; alternating broad cathode and narrow anode strips are etched in a thin ( ~ 1 Ixm) metallised layer on a substrate. The narrow anodes act like wires in a conventional chamber, providing gas amplification for electrons drifting in from the gas volume. The strong dipole field between anode and cathode quickly clears the positive ions from the anode region, producing a fast signal. One problem suffered by MSGCs is that the highest gain that can be reached before breakdown of the gas in the anode-cathode gap is about 103 to 104. At these gains it is hard to detect the signal from a single primary ionisation in the gas and this may limit the efficiency of the thin gas-gap MSGCs which are needed for high rate applications. When a MSGC is run near its gain limit there are occasional sparks which can erode the electrodes (Fig. 1). This figure immediately shows that the breakdown is localised at the ends of the cathodes, which is also the region where the electric field near the cathode surface is at its maximum. This maximum gain is influenced by the gas mixture used [3], the substrate type (thin layer or bulk conductivity) [4] and the thickness [5] and shape of the electrodes. We have chosen to concentrate on optimising the shape of the cathode ends because this does not add to the cost and does not influence the other performance characteristics of the detector. We have used a 3-D simulation of the electrostatic field and a 40 X life-size scale model to understand the breakdown mechanism and find an improved shape for the cathode ends. The reference design, which we hope to improve on, is
an MSGC with 100 p.m wide cathodes and 10 p~m wide anodes at a pitch of 300 p.m made from 1 p.m thick aluminium. The cathodes have semi-circular ends. The substrate has some bulk conductivity sufficient to prevent charging up and we believe that we can ignore the different dielectric permittivity of gas and substrate because we operate with a field configuration in which few field lines cross the gas-substrate surface. The gas is a r g o n / i s o b u t a n e 7 5 / 2 5 and we know that a real chamber of this design has a gain of about 3000 at 700 V and it breaks down at about 720 V.
2. Electrostatic simulation We have used a multi-particle approach based on the ideas of ref. [6] and extended it to describe many conductors at different voltages. First we describe our algorithm in some detail while the next section covers the use and interpretation of our final electric field maps.
* Corresponding author. 0168-9002/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0 1 6 8 - 9 0 0 2 ( 9 4 ) 0 0 5 3 0 - K
Fig. 1. Damage caused by sparks to the electrodes of a MSGC.
L Duerdoth et al./Nucl. Instr. and Meth. in Phys. Res. A 348 (1994) 356-360
357
2.1. Method This simulation only describes a system of perfect conductors surrounded by a uniform dielectric. We will use the indices k and l to label conductors and i and j to label particles. Suppose that we wish the conductors to be at potentials Vk. Each conductor is populated with a number of identical point particles of charge Q~. For a given distribution of particles the following quantities are calculated: is the vector from particle i to particle j. eij = Ei ÷ j Q i / I Rij I is the potential energy of a unit charge at the position of particle j and Ey = QjPy is the potential energy of the particle itself. Fj = ~]i ÷ j Q i a j R i j / [ Riy [ 3 is the force on the particle j. Vffv is the average value of Py of the charges in conductor k. VRMs is the root mean square deviation of the Pjs from the desired voltage of their conductor, Vk. We normalise V~ Ms to the scale of the largest voltage difference in the system, in this case 700 V. With this information availible, one of three possible steps can be used to converge towards the desired voltages (i.e. to minimise vRMS). 1) Move each particle some distance in the direction of the force Fj. The distance is chosen to be some small fraction f times the distance of the particle from its nearest neighbour. When a particle comes up against the surface of the conductor it is moved in the direction of the component of F which is perpendicular to the surface. The fraction f is initially 0.1 and is reduced as the system nears equilibrium. 2) Move just one particle in each conductor. Choose the particle with the highest value of Ej in each conductor and place it near to the one with the lowest Ej in the same conductor. 3) Change the charges of all of the particles such that the mean potential in each conductor VAv is at the desired value Vk. This can be done by computing the capacitance matrix Ckt = ~i~.j~:i 1 / I Rij l, where index i runs over all charges in conductor k and index j runs over all charges in conductor 1. The inverse of Ckt multiplied into V~ gives the new values of the charges for each conductor. The initial positions of the particles are randomly distributed within the volume of their associated conductors and ~ 1000 iterations of the above steps are used to reach a solution. Most of the work is done with iterations of type 1' and 3 but type 2 is useful at the early stages for removing long-range variations of potential which take a long time to smooth out with type 1 steps. Steps of type 1 and 3 will only converge towards the desired solution if the diagonal terms of the capacitance matrix are large compared to the off-diagonals. This condition can be achieved by increasing the number of charges on the conductors.
f
•
I
.t
q
•
I
•
I
•
I
+ o
I,
Fig. 2. The final distribution of charges in our simulation of a MSGC of conventional geometry. The points give the positions of the charges and the thin lines are the boundaries of the conductors.
Another trick that we have used is to start out with a relatively small number of particles ( ~ 50) in each conductor and when these have reached equilibrium to double their number by placing a new particle in the space between each particle and its neighbour. This way the new particles start out from a position which is close to equilibrium over long ranges and only local rearrangements are needed to get back to a good solution. In addition, one physical conductor may be split up into a number of smaller parts for the purposes of the simulation and the parts which we are interested in may be simulated in greater detail than the rest. The quality of the electric field solution is indicated by V RMs. The results of the next section are derived from a distribution of 5800 particles in the region of interest and 700 particles on the surrounding conductors which is shown in Fig. 2. This simulation used around 30 hours cpu time on a HP735 (40 Mflops). The v R M S s of the 14 conductors are in the range 1-3%. While searching for an improved shape of cathode ends we have used a similar amount of computer time to produce field strength maps for a more complicated arrangement of conductors such as Fig. 5, in which the V RMs values of the important conductors were in the range 3-7%. II. MICROSTRIP GAS CHAMBERS
358
L Duerdoth et al. /Nucl. Instr. and Meth. in Phys. Res. A 348 (1994) 356-360
2.2. Results
Fig. 3 shows a contour map of the electric field strength near the end of a cathode strip of the standard design. We are looking in the plane parallel to the conductors but 5 Ixm inside the gas gap. The contour interval is 1 V p,m-1 • The highest field near the tip of the cathode is 16.8 V Ixm- 1 while the field near the straight side of the cathode is 10.7 V txm -1. We believe that it is these high fields which are causing the breakdown because an electron liberated from the surface of the cathode will start a significant avalanche within 10 Ixm and photon feedback cannot be absorbed within this short distance and will liberate another electron. This is investigated by looking at examples of field lines originating at the tip and edge of the cathode and indicated by plain and dashed lines on Fig. 3 respectively. We chose field lines which leave the cathode at an angle of about 45 ° to the plane of the conductors and loop out into the gas volume before returning to the anode. Fig. 4a shows the field strength versus the distance along the field lines while Fig. 4b shows the value of the first Townsend coefficient, a , corresponding to these fields• The Townsend coefficients were computed with the program M A G B O L T Z [7] and are only approximate for
/ \
/ Fig. 3. Contours of equal electric field strength in the plane inside the gas of an MSGC 5 p~m above the electrodes. The contour spacing is 1 V p,m-1. The plain and the dashed lines are field lines from the tip and edge of the cathode respectively.
o ._o 3O
l
(a)
i
I
E ~ 20
~-o 10 ~z 0
i
J 0
20
40
60
80
1O0
t 20
Distance (microns) ~ 1.5 L E ~ t "~ .c ~" 0,5 ~,_ 0 c ~o J-
o
15
(b)
20
:_
(c)
0
20
i
40
60
80
1O0 120 Distance (microns)
_J
,-° o. -6 10 "_o ~ 5 -¢
0
, , , I , , ,
40
60
80
100
120
Distance (microns)
Fig. 4. (a) The field strength plotted as a function of the distance along the two example field lines from the cathode end (plain) and the cathode edge (dashed). (b) The corresponding Townsend first coefficient. (c) The integral of the first Townsend coefficient. fields greater than 10 V p,m -1 because good data for isobutane is not available. However, we have found that the relative gains in different parts of the chamber have little sensitivity to the shape of the a ( E ) curve within the range of variation that we have studied. Fig. 4c shows the integral of the Townsend coefficient along the field lines. One can see from Fig. 4c that an electron drifting in from the low field region to a typical part of the anode (right hand half of the dashed line) will be amplified by a factor e 9. This agrees reasonably well with the gain of e 8 = 3000 measured with the real detector and gives us confidence that the other gains predicted by this simulation will be correct within an order of magnitude. For example we predict that in the first 10 p~m of travel away from the cathode surface an electron will produce an avalanche of e 72 = 1300 electrons if it came from the tip of the cathode or e 43 = 74 electrons if it came from the straight edge of the cathode.
3. I m p r o v e d d e s i g n
Our aim is to increase the gain that an MSGC can reach before breakdown and it is clear that one way to do this is
L Duerdoth et al./NucL Instr. and Meth. in Phys. Res. A 348 (1994) 356-360
359
4. Scale model
( ! • i • ! • it * •
, •
°
, o o ° *
° ° i
•
• o *
i i
Fig. 5. An example of charge distribution on the improved cathode design which slightly lowers the maximum electric field near the cathode surface.
to reduce the maximum field that occurs near the cathode surface. This could be done by thickening and rounding off the edges of the cathodes, which would require a change to our etching method, or alternatively by changing the mask pattern. We have changed the pattern by terminating the cathodes with " b l o b s " which have larger diameter than the cathode width in the working part of the detector (Fig. 5). In order to leave space for the anodes, alternate cathodes must be given a neck and extended beyond the other cathodes. We have found that the amount of improvement possible from this scheme is so small that it cannot be comfortably measured when the accuracy of our simulation is only + 3%. The simulation did tell us though that the field at the tip of the short cathodes can be reduced by increasing the size of blobs on the long cathodes, because the repulsion of these two areas of negative charge tends to prevent charge from accumulating at the tip of the short cathode. The difficulty of making a sufficiently accurate simulation persuaded us to go to the scale model described in the next section.
The properties of a gas relating to its breakdown under an applied electric field are always functions of the ratio of field strength to gas pressure. This means that we can scale the dimensions of our MSGC up by some factor and the gas pressure down by the same factor and expect that its behavior will be unchanged. The one thing that is changed by this scaling is the time duration of the processes. If there was an important fixed time scale involved in breakdown, for example the lifetime of a metastable state, then our scale model would not reproduce the real chamber. We have used a scale factor of 40 which enables us to build test structures easily on printed circuit board and test them in a low pressure chamber of reasonable dimensions (20 × 20 cm) with transparent windows. The breakdown voltage was measured by turning up the applied voltage slowly and recording the point at which a visible discharge appeared. This was repeated several times to obtain the error estimate. The scale model of our standard design breaks down at 915 + 10 V at the cathode ends whereas the real chamber breaks down in the same place at about 720 V, showing that the scaling law works moderately well. Many other test structures have been tried, all following the theme of "alternating blobs" which was described above. The overall picture is that it is hard to find a structure which is any better than the plain semi-circular ends for terminating the cathodes. However, we have found one pattern (Fig. 5) which is slightly better, holding up to 940___ 10 V before breakdown. The " r u l e s " of a better pattern seem to be that the anode-cathode gap must always be larger than the one in the sensitive part of the detector, the cathode should not be necked down below about 50% of its width and the blobs on the longer cathodes should be as large as possible, even though it is not here that the breakdown occurs.
5. Conclusion A multi-particle method of simulating the 3-D electrostatic field around a fairly complex group of conductors at chosen voltages is described. This simulation can produce a field map accurate to about 2% in the space between the conductors, provided that one does not look too close to the conductors where the influence of individual point charges becomes noticeable. Combining our field map with various parametrisations of the first Townsend coefficient has shown that when the amplification at the anodes is at a useful value ( ~ 10 000) then there is inevitably an amplification of order 1000 within 10 p~m felt by an electron leaving the cathode tip. A small ( ~ 3%) increase in the maximum sustainable voltage can be achieved by changing the shape of the cathode II. MICROSTRIP GAS CHAMBERS
360
L Duerdoth et aL /Nucl. Instr. and Meth. in Phys. Res. A 348 (1994) 356-360
ends. This leads to a worthwhile increase of about 50% in the m a x i m u m gain o f the chamber.
References [1] RD28 Status Report, CERN/DRDC/93-34. [2] A. Oed, Nucl. Instr. and Meth. A 263 (1988) 351.
[3] M. Geijberts et al., Nucl. Instr. and Meth. A 313 (1992) 377. [4] F. Angelini et al., INFN P I / A E 93/07 (1993), submitted to Nucl. Instr. and Meth. [5] Yu. Pestov and L. Shekhtman, BINP 93-59 (1993). [6] H. Van der Graaf and J.P. Wagenaar, Nucl. Instr. and Meth. 217 (1983) 330 and A 252 (1986) 311. [7] S. Biagi, Nucl. Instr. and Meth. A 283 (1989) 716.