Signal development and processing in multi wire proportional chambers

Signal Development

and Processing

in

Multi Wire Proportional Chambers

Signaal ontwikkeling en -verwerking

in proportionele dradenkamers

P r o e f s c h r i f t

ter verkrijging van de graad van doctor in de technische wetenschappen aan de Technische Hogeschool Delft, op gezag van de Rector Magnificus prof. dr. J.M. Dirken, in

het openbaar te verdedigen ten overstaan van het College van Dekanen op donderdag

19 juni te 16.00 uur door

Hendrik van der Graaf

natuurkundig ingenieur, geboren te Rockanje

TR diss 1496

Dit proefschrift is goedgekeurd door de promotor

prof. dr. A.H. Wapstra

2 mm

X-ray shadow image of a fish hook recorded using a Multi Wire Proportional Chamber. Irradiation with 6 keV quanta from a 10 mCi

55Fe source placed at a distance of 0.7 meter. Exposure time: 4 hours.

DIMDEX

Introduction 5 Conventions and nomenclature 9

Chapter 1. The Multi Wire Proportional Chamber

as localizing X-ray detector 11

1.1 Principle of operation 11 1.2 The interaction of X-ray quanta with gases 12

1.2.1 Absorption of X-ray quanta in a gas layer 13

1.2.2 Fast electrons in gases 14 1.2.3 Fluorescent X-rays 19 1.3 Drift of electrons in gases 20 1.4 Gas multiplication 21 1.5 Applications 22

1.5.1 High-energy physics 22 1.5.2 Nuclear physics 24 1.5.3 MWPCs as a substitute for photographic film 24

Medical applications 24 X-ray diffraction 25 Astrophysics 25 Chapter 2. Induced charge on the electrodes of a

MWPC 27 2.1 Introduction 27 2.2 Reprint: A calculation in three dimensions of

the induced charge on the electrodes of a MWPC 29 2.3 Preprint: Numerica! results of calculations in

three dimensions of the induced charge in MWPCs 33 2.4 Additional measurements of induced charge

distributicns on wires 53 2.5 Preprint: The L3 high-resolution muon drift

chambers: systematic errors in track position

measurements 55 2.6 The drift velocity of ions in gases 67

2.7 Reprint: Signal development in wire chambers: the

drift velocity of ions in gases 69

Chapter 3. Error sources 75 3.1 Inclined radiation 75 3.2 Non-homogeneous electrical field 75

3.3 Amplifier noise 77 3.4 Finite path length of photoelectron 80

3.5 Multiple X-ray conversion points 80

3.6 Multiple avalanches 81 3.6.1 Cell-crossing tracks 81 3.6.2 Secondary avalanches initiated by UV light 81

3.7 Avalanche spread over the surface of the wire 82

3.8 Electron and ion diffusion 82 3.9 Mechanical tolerances 82 3.10 Summary of error sources 83 Chapfer 4. The Test Chambers 85

4.1 A modular chamber 85 4.2 A high-pressure chamber 85 4.3 A high-pressure, low-absorption window 87

Chapter 5. A multi-ADC readout system 91 5.1 Induced charge signals on cathode planes

and adjacent wires 91 5.2 Induced charge on wires: the Y-coordinate 93

5.3 Induced charge on strips: the X-coordinate 94

5.4 X-ray shadow images 96 Chapter 6. A fast readout system 99

6.1 Introduction 99 6.2 Signal selector: analog switches 99

6.3 An analog signal processor 99 6.4 The digital ratio convertor 101 6.5 Reprint: A novel, two-dimensional, fast, low-cost

and accurate readout system for MWPCs 106 6.6 Additional remarks on the fast readout system 109

6.6.1 Switch unit 109 6.6.2 Signal processor 111 6.6.3 Trigger 114 6.6.4 Spatial resolution 116 6.6.5 Processing speed 118 6.6.6 Calibration procedure 119 6.6.7 Multiple avalanche error 120 6.6.8 Error due to digitizing 121 Chapter 7. Improvements 123

7.1 Alternating sense-field wire geometry 123

7.2 Top-bottom strip coupling 123

7.3 PROMvetoes 123 2

7.4 7.5 7.6

Multiple track recognition

Improving Y-coordinate spatial resolution by measuring the ratio QUp / Q , j0 w n

Improving the readout system Symmetrical inputs and outputs Hybrid preamps

Discrete FET's as switches FADC buffers

Automatic calibration Time constant of the charge sensitive preamp Conclusions Literat Summ ure ary Samenvatting Nawoord Curriculum Vitae 124 124 124 124 125 125 125 125 126 127 129 133 134 137 139 3

Fig. 1.1 The Multi Wire Proportional Chamber (MWPC): a plane of parallel and equidistant wires is sandwiched between two cathode planes, consisting of strips which are insulated from each other. The strips are perpendicular to the wires in at least one cathode plane.

4

IntroductSoo

In 1978 Charpak, Petersen, Policarpo and Sauli irradiated a Multi Wire Proportional Chamber (MWPC) with soft X-ray quanta. They analyzed the charge signals from wires and cathode planes simultaneously for each quantum that interacted with the gas in the chamber.

These measurements confirmed the existing model for charge signal development of proportional counters: the X-ray quanta create tiny clouds of free electron-ion pairs in the chamber. The free electrons drift along a field line to the nearest anode wire, causing an avalanche. When the avalanche is complete the electrons created during the avalanche have arrived at the wire. The ions, however, with their much smaller drift velocity, are still close to the wire surface, drifting along their local field lines towards the cathodes. A positive charge is therefore induced on the wires and cathode planes. The charge on the avalanche wire is the sum of a) the negative electron charge of the avalanche electrons arriving at the wire and b) the positive induced charge caused by the ions. Just after the completion of the avalanche these two charges cancel each other out. The time development of the charges on wires and cathodes is determined by the movement of the ion cloud.

Charpak and his group observed differences between the charge signals from the two wires adjacent to the avalanche wire; also, the signals from the top and bottom cathode planes were not equal. A dependency was found between these differences and the point of absorption of the X-ray quantum. This dependency could only be explained by the fact that at moderate gas amplification the avalanche is limited to a specific spot on the wire surface.

In the Geiger mode the avalanche is propagated along the entire length of the wire owing to space charge effects and UV light propagation. In the proportional mode the avalanche was assumed to spread around a limited length of the wire. This spread appeared to be much smaller: the ions are located mainly at the point of arrival of the electrons. The ions drift back along the same field line as that along which the electrons drifted towards the anode wire.

The limited geometrical spread of an avalanche has also been observed by Fisher, Okuno and Walenta.

The cathode planes of Charpak's chamber were made of strips insulated from each other. By measuring the charges on individual wires and strips Charpak and his group showed that they could measure two coordinates of X-ray quanta with a precision which was limited only by the finite range of the photoelectron created in the photon absorption process.

Our work involved studying the possibilities of using the MWPC tor X-ray deteotion, for instance as a substitute for photographic film. For this application the quality of the device depends on its spatial resolution, efficiency and counting rate.

The efficiency of an X-ray sensitive MWPC is determined mainly by the probability of absorption of X-ray quanta by the gas in the chamber, Where absorption takes place a cloud of free electron-ion pairs is created. The average distance between the point of absorption and the centre of gravity of the electron cloud determines the spatial resolution. The interaction of X-ray quanta with gases was therefore studied in detail: the relevant quantities are presented in graphs.

A MWPC used as an X-ray localizing detector may require a high gas pressure. A window was therefore developed to withstand this pressure; a composite construction using carbon fibre as material keeps the absorption of X-ray quanta by the window itself fairly low.

The charge signals from a MWPC initiated by the absorption of each quantum have to be processed individually. The signals have to be converted by a processor into digital X and Y-coordinates for the point of absorption. To limit the exposure time the counting rate of the processor has to be very high. The present thesis describes a processor which is essentially capable of handling counting rates of up to 5 . 106 events per second.

To design this processor a good knowledge of the charge signal distribution and development was required. The calculation of these signals can be reduced to computing the induced charge on grounded conductors due to the presence of a point-like space charge somewhere between them.

In recent years there have been many attempts to solve this problem in relation to the geometry of MWPCs. The present thesis describes a method which is independent of the geometry of the conductors. The accuracy of the method depends only on the available computer power. The algorithm is highly suited to the architecture of 'long pipeline' vector processors. A Cyber 205 Supercomputer was used to compute numerical values for our particular chamber geometry. Parameters such as wire spacing and space charge position were varied; in this way some general relationships were ascertained.

The results of the calculations of charge distributions are compared with measurements using two MWPCs and a special proportional tube with a very small cathode radius.

6

The MWPC was originally used to localize tracks of charged particles. The only difference between the interaction of X-ray quanta and that of charged particles is the geometrical spread of the pnmary electron-ion pairs. The signal development of MWPCs can be descrlbed in ferms of superposition of the contributions of each individual free electron that arrivés in the gas amplification region near the avalanche wire. For this rcason a more gcncral titlo was chosen for this thesis.

Chapter 1 sets out the principle of the MWPC as an X-ray localizing detector. The interaction processes between X-ray quanta and gases are described. Theories of electron drift and gas multiplication are mer.tioned briefly; the values of the relevant parameters of these two processes are such that a simple model is adequate.

The last part of Chapter 1 presents some applications of the chambers and processors described in the thesis.

Chapter 2 describes the calculation of the induced charge in MWPCs. The algorithm used on the Cyber 205 Supercomputer was also used to calculate the signal development in the L3 high-precision drift chambers. The result predicts some effects which occur in the measurements of track slopes. Measurements of signal development in the narrow proportional tube are presented.

Chapter 3 discusses the processes which cause deviations between the mcasurcd and rcal point of absorption of an X ray quantum. This study assumed the use, where approriate, of the readout system described in Chapter 6.

The design and construction of the chambers is set out in Chapter 4, where the construction of the high pressure window is also discussed.

Chapter 5 describes the measurements using a MWPC and 24 ADC channels. The calibration procedure and some X-ray images are presented.

Chapter 6 explains the rapid readout system. Chapter 7 lists possible improvements to chambers and readout systems.

* \ oj t •

t f

Q

Q

Fig. 1.2 Quaütative impression of charge distributions over the electrodes ofa MWPC shortly alter the completion of an avalanche. Three cross-sections are shown. The arrow length is proportional to the charge signal. An up-arrow indicates a positive charge as measured by the outside world. a down-arrow a negative charge. Names of charge signals are indicated.

8

CQNVENTONS AND NOMENCLATURE.

1. The temperature is 293 (5) K, unless otherwise stated.

2. The Standard chamber has a gas gap of 10 mm, a wire spacing of 2 mm and a wire diameter of 20 urn. Angle 9 = 0° See fig 1 1 for the coordinate system. In the present thesis we refer to this chamber unless otherwise stated.

3. Names of charge signals and other parameters are indicated in

4. A tast electron is a free electron with an energy high enouqh to create lon-electron pairs.

5. Primary or free electrons are created, together with ions by a fast electron travelling through gas.

6. Resolutions are given as 0, unless otherwise stated. 7. Fig.6.7 means figure 7 in Chapter 6

8. References to literature are given in square brackets, e.q r201 The literature is listed on page 129. l J

References to the text are given in round brackets, e q (4 2) the figures refer to chapter and section.

Formulas used in this thesis are labelled as {5.4}; the first fiqure

reters to the chapter where the formula is first used.

9. Figure and literature references in reprints or preprints relate to the paper in question.

10

' L ^

Rnlo»

s y

range

11.

of 5 - 50 keV.

The PROM square is a plot of a ratio of two charge signals against another ratio of two charge signals. Both ratios are functions of the same parameter. In a practical situation both ratios appear as a 6-bit digital word. The twelve-bit combination of these words forms an adress for a 4096*8 Programmable Read Only Memory (PROM) The output of the PROM is the value of the parameter. The two ratios are mter-dependent; this is represented by a curve in the plot.

,HMim

9 n

12. A chamber cell is the volume in which an ion or electron travels to a particular cathode strip or anode wire. Each strip/wire determines a cell. We refer to X and Y cells, defined by strips and wires respectively.

Fig. 1.3 Electrical field Unes in a quarter cross-section of a Ycell of a MWPC. An electron drifting along the indicated field line arrivés at the wire with incident angle f.

10

Chapter 1

The fMylti Wire Proportional

Chamber as localizing X-ray

detector

1.1 Principle of operation

Two-dimensional positional information on X-ray quanta can be obtained using a chamber as shown in fig.1.2, consisting of a plane of equidistant thin metal wires between two cathode planes parallel to the wire plane. One cathode plane is segmented into strips of equal width insulated from each other. The strips are perpendicular to the wires. The gap between the cathodes is filled with a suitable gas. The wires are held at a high positive electric potential in relation to the cathode strips.

Assume that an incoming X-ray quantum has a direction perpendicular to the wire and cathode planes of the chamber. The quantum may interact with the gas in the chamber; this process depends on the quantum energy and the gas mixture. In the case of quanta with an energy below 50 keV photoelectric absorption is the most probable process. The photoclectron is released from a gas atom or gas molecule; its energy equals the quantum energy minus the binding energy of the electron. The ionized atom or molecule may emit one or more secondary X-ray quanta or an Auger electron. In travelling through the gas the photoelectron will lose energy as described in Bethe-Bloch's theory. Part of this energy loss causes ionization of atoms and molecules, which results in a cloud of free low-energy electron-ion pairs. Let us assume that the range of the photoelectron is fairly small compared to the wire spacing.

An electron in the cloud drifts along the local field line to the anode wire. The average velocity of each individual electron is determined by the electrical field, the mass of the electron and collision processes between the electron and atoms or molecules.

The local electrical field increases as the electron drifts closer to the wire. Near the wire surface the field is so strong that the energy increase of the electron between two collisions with atoms or molecules allows the electron to ionize an atom or molecule; thus a single free electron can create an avalanche of a million or so electrons on the surface of the wire. This gas multiplication process allows us to detect an electronic pulse at the wire. Since the drift velocity of electrons is about 1000 times larger than that of the ions, the avalanche leaves behind a positive ion cloud

whose density increases towards the wire. The negative electron charge created in the avalanche first stays in the wire close to the positive charge. Due to the electrical field the positive charge slowly drifts towards the cathode, following the same field line as the previous free electrons. The negative electron charge on the avalanche wire flows away. Nearby strips and wires 'see' a free positive charge and therefore rise in potential. Electrons flow from ground into these electrodes.

We define the 9 as the angle between the wire plane and the field line along which the ion cloud moves.

Suppose we measure the total charge flow from the wires and strips during a short fixed time interval after the moment of avalanche. Since the electrical field is almost radial close to the wire, the positive charge is at a fixed distance from the avalanche wire, for each quantum, at the moment that the measurement is completed. This distance does not depend on angle <f .

Fig.1.2 shows a typical charge distribution over the chamber electrodes in this situation. The distribution depends only on the chamber geometry and the position of the positive charge; it can be described in terms of electrostatics since the ion drift velocity is very low compared with the speed of light. Thus, the ion space-time relationship determines the signal development at the electrodes.

The distribution over the strips obviously carries information on the X-coordinate of the quantum: the bell-shaped symmetrical distribution peaks at the avalanche position.

Fig.1.3 shows a cross-section of the chamber perpendicular to the wires. The ratio between the induced charges on the adjacent wires is a measure for the angle <p . Since the electrical field is homogeneous in the larger part of the chamber, <p gives a good value for the Y-coordinate for most X-ray interactions. The ratio of the charges on the adjacent wires is shown for different values of <p in fig.1.4. The Y-coordinate is therefore a function of the charge distribution over the wires.

The signals from the strips and wires of a MWPC thus carry information about the field line on which the quantum has created a fast electron. The major part of this field line has a specific X and Y-coordinate, related to the point of absorption of the X-ray quantum.

1.2 The interaction of X-ray quanta with gases

The performance of an X-ray imaging MWPC is characterized mainly by its spatial resolution and efficiency. The following quantities are important:

12

1.4

0 25 50 75 100 125 150 175

* (DEG)

Fig.1.4 Agraph of the ratio 0-\/Q.-j as a function off.

- the probability for incident X-ray quanta being absorbed in the gas layer of the chamber;

- the range of the photoelectron and -if emitted- the Auger electron; and

- the radiation length of the gas in the case of fluorescent X-rays. 1.2.1 Absorption of X-ray quanta in a gas layer

In our work the X-ray energy used was only 6 keV: we are concerned solely with photoelectric absorption. For photographic film, for instance, the X-ray energies may be much higher. Even then, if the chamber gas is a heavy inert gas, photoelectric absorption is the main interaction process [1]. This quantum process removes one or more electrons from the electron shells of an atom or molecule. The quantum energy E3 must then be larger

than the shell energy Ex. The quantum is totally absorbed and a

photoelectron of energy E^ - Ex is emitted. The hole thus caused in the

shell can be filled again by:

- fluorescence: an electron transition from a higher shell j into shell i with a missing electron. A photon of energy Ej - E; is emitted.

- the Auger effect: a non-radiative internal rearrangement involving several electrons from the higher shells. This results in the emission of an electron with an energy close to Ex.

o

d

1.0

0.7

The absorption probabilities ( nk, |i.| etc) in different shells are

superimposed to give a total interaction probability. The probability of absorption by a shell is maximum just over the shell's energy, decreasing rapidly with quantum energy. In the oase of quantum energies below a few MeV the absorption coëfficiënt of a particular shell can be expressed as:

H - Z». E. -3.5 {1.1}

Figs.1.5-1.7 show the absorption in a gas gap with Ar, Kr and Xe gas at normal temperature and different pressures as a function of the quantum energy [2], The presence of a quenching gas contributes to absorption; any calculation of the efficiency of the chamber should take this into account.

The gap of 10 mm is Standard throughout the present thesis. The

size of the gap is a compromise: a wider gap would result in a more efficiënt X-ray detector but spatial resolution may be impaired by parallax error in the event of non-perpendicular incident radiation.

1.2.2 Fast electrons in gases

The emission of a photoelectron or Auger electron -hereinafter referred

to as a fast electron- results mainly, according Bethe-Bloch's theory, in a

X - R A Y E N E R G Y ( k e V )

Fig.1.5 Absorption of X-ray quanta in WmmArgas.

14

X

A B S O R P S I O N O F X-RAY Q U A N T A IN 1 0 M M L A T E R Kr GAS

l-RAY ENERGY ( k

Fig. 1.6 Absorption of X-ray quanta in 10 mm Kr gas.

ABSORPSION OF X-R4Y QUANTA IN 10MM LAYER Xe GAS

X-RAY ENERGY

Fig. 1.7 Absorption of X-ray quanta in 10 mm Xe gas.

GAS LAYER

Fig. 1.8

The path of a last electron. lonization increases at the end of the path. The lower curve shows the resutting density distribution of ion-electron cloud. Note that for this particular path the X-coordinate of the centre of gravity of the cloud is approx. equal to the X-coordinate of the point of absorptlon of the quantum.

Fig. 1.9

Polar plot for the probabillty of emission of a photoelectron at a glven angle between incoming photon directton and the direction of emission, as a function of the quantum energy

cloud of electron-ion pairs. What we can measure are two coordinates of the centre of gravity of the electron cloud. However, this point is not the interaction point of the X-ray quantum. The distance between the centre of gravity of the free electron cloud and the interaction point of the quantum causes an error. This distance depends on:

- The energy of the fast electron,

- The energy loss, and proportional to that, the ionization per unit length, - The angle of emission of the photoelectron, and

- The scattering of the electron.

Suppose a quantum arrivés along the Z-axis (fig.1.8). At x=y=z=0 photoabsorption takes place; a photoelectron is emitted. In fig.1.9 the probabilities are shown for the angle between the photon direction and the direction of the emitted photoelectron. Note that of low photon energies the preferred direction for the photoelectron is in the X-Y plane of the ch amber.

16

Now the electron creates electron-ion pairs along its track. In most gases it loses about 23 eV for each electron-ion pair created [3,4]. The energy loss per unit of length is described by the Bethe-Bloch theory; a qualitative indication for energies up to a few keV is:

dE __ J_

dx = f {1.2}

where £ » v/c; v is the electron speed [3]; g is the density of the gas. lonization peaks at the end of the track. We define the relative centre of gravity of the free electron cloud as:

3 = ± ƒ Rp(R) dV - P {1 3j

where R(x,y,z) is the position vector of the charge deposit P dx.dy.dz, Q the total lonization:

and P the position vector of the interaction point, Integration yields the volume of a sphere with a radius larger than the maximum range and P- as centre of the sphere.

Equation {1.3} is difficult to resolve since the fast electron is scattered many times. Owing to the low mass of the electron the path is randomized. A Monte Carlo computation can provide the most probable values and spread in S as a function of gas mixture, gas pressure, chamber geometry and electron energy.

Only the X and Y-coordinates of absorbed quanta are of interest: for this reason the Z-component of 8 is not relevant. In effect we are dealing with the projection of G on the X-Y plane of the chamber.

For the present work we estimated iSl as the electron range defined by Katz and Penfold [5,6]. This quantity is defined in fig.1.10. It shows the relative number of fast electrons reaching a counter after passing an absorber as a function of the thickness of the absorber. Extrapolation of this curve and the background curve identifies the practical range. From many measurements an empirical formula was derived:

Rp = 0.412 E<1-2 6 5-°-0954 In E)

for 0.01 i E < 2.5 MeV

E L E C T R O N RANGE IN A r

TRANSMISSION OF MONO'ENERGF TIC ELECTRONS

BSORBHON THICKNESS ( g / c m ' ) *■

ELECTRON BEA

Fig.1.10

Definition by Katz and Penfoid of the range of a fast electron; extrapolation of absorption and background curves.

Fig.1.11

E L E C T R O N ENERGY f k e V ) — Range of a last electron in Ar as a function ofitsenergy.

ELECTRON RANGE in Y ELECTRON RANGE

Fig.1.12 Fig.1.13

ELECTRON ENERGY (keV)- ELECTRON ENERGY (keV)

Range of a fast electron in Kr as a function Range of a fast electron in Xe as a function cf its energy. °< *« energy.

18

where R „ is the practical range, expressed in gram cm"2 and E the electron energy in MeV.

Bateman and others [7,8] inserted a t e r m taking into account the screening effect of higher Z materials:

Rp = 0 , 2 1 1 5 Z ° -2 6E (1-2 6 5- 0 - 09 5 4 l nE ) {1.6} In the case^of real tracks as shown in fig.1.8 we find a significantly lower value for G compared with the practical range according to Katz and Penfold. The latter is a measure of the range in the case of a rather straight line as the path. Thus the calculated practical range is too large. In the present thesis we shall take expression {1.6} to be a conservative estimate ot the distance between the point of emission of the fast eiectron and the centre of gravity of the electron cloud.

In figs. 1.11-13 the ranges are shown as a function of the energy of the electron and gas pressures with different gases.

With low-Z materials the emission of an Auger electron is favoured: see fig.1.14. The angle of emission is not correlated to the direction of the X-ray quantum. The average projection of vector G on the X-Y plane of the charnber is therefore smaller in the case of Auger electrons. We shall however treat photoelectrons and Auger electrons in the same way in this respect. In this way we somewhat overestimate the uncertainty in measured quantum coordinates in the case of emission of Auger electrons.

Fig.1.14

Probability of emission of a fluorescent X-ray as a function of atomic number Z. Note that with Ar (Z=18) an Auger electron is emitted in 85 percent of the interactions. Z^r = 36, Z^e = 54.

1.2.3 Fluorescent X-rays

An emitted fluorescent X-ray may be absorbed far away from the primary interaction point. The electron-ion pairs j;reated by this secondary photo absorption change the position vector G of the centre of gravity of the total electron cloud. The magnitude of this extra deviation depends on the energy of the fluorescent quantum, since the charge of the additional

electron cloud is proportional to the energy of the secondary photo electron. The range of this electron also depends on this energy as before. The mean free path of the quantum decreases with gas pressure as shown in fig.1.15 for Ar, Kr and Xe.

10

MEAN FREE PATH FOR FLUORESCENT X-RAY'S FL. X-RAY ENERGIES A r KT X e 3,2 keV 12.6 keV 29.0 keV GAS PRESSURE ( B A R )

Fig. 1.15 Mean free path for fluorescent X-rays as a function of gas pressure. Note that with 10 bar of Ar this value is 10 mm. Many of these secondary X-rays wil\ therefore not escape, thus causing multiple avalanches.

1.3 Drift of electrons in gases

A cloud consisting of free electrons drifts along the local field line towards a wire. The drift speed of electrons does not play a role in this method of localizing and will not be discussed here.

A drifting electron collides many times with atoms or molecules and does 20

not follow one field line. The charge cloud diffuses; we can estimate the size of the cloud at the moment of arrival in the multiplication region [4]. We assume the cloud of free electrons to be a point at the moment of creation. The width of the distribution is expressed as an r.m.s. value and written as:

Ö = D / L {1.7} where D is the diffusion constant, depending on gas mixture and pressure

and electrical field. This quantity is often expressed in u.m c m " "2. L is the drift path length of the electron cloud. In the Standard chamber the maximum value of L is 6 mm. There are many experimental values for D, since this quantity is important in high-accuracy drift chambers [9]. A recent value for D measured by Hartjes and Konijn [10] with a mixture of 30-70 percent ethane-Ar gas is:

D = 157 p.m c r r f1 / 2

With a drift length of 10 mm the electron cloud has a spread of 157 nm. The uncertainty of the position of the centre of the cloud will depend on the number of electrons in the cloud. Since this figure is in the order of 200 or more we may assume that the centre of gravity is accurate within

10 urn.

1.4 Gas Multiplication

Close to the wire the electrons are accelerated so strongly that their energy increase between two collisions is sufficiënt to ionize an atom or molecule. This effect is discussed in detail in [4], [11], [12], [13].

An important quantity for gas multiplication is the mean free path for ionization. The value depends on the gas mixture and pressure and decreases with increasing local electrical field. In a proportional chamber the mean free path for ionization has the lowest value at the surface of the wire; the value here is in the order of one micron. The average ion density increases by a factor of two over the length of one mean free path. This means that in the case of negligible spread around the wire, the centre of gravity of the ion cloud created in the avalanche is only two or three microns from the wire surface. According to recent measurements [10], it appears that the drift velocity of electrons in the very strong electrical field near the anode wires is higher than 100 nm/ns. Since the gas multiplication region starts in the order of 25 u.m from the surface of the wire the m u l t i p l i c a t i o n

JJUUHfUU

p r o c e s s d u e t o o n e p r i m a r y e l e c t r o n is c o m p l e t e d w i t h i n a f r a c t i o n of a n a n o s e c o n d .

Interesting results from a Monte Carlo simulation of an avalanche were published recently by Matoba and others [12], It appears that at moderate gas gains the avalanche has a non-negligible spread in the X and f -coordinates over the surface of the wire. This is caused by the wild movement of the free electrons in the strong electrical field and by the change in the electrical field near the wire due to the build-up of positive space charge. The propagation of UV photons emitted by argon ions may also contribute to this effect.

A value for the FWHM of the <p-coordinate is about 4 0 ° for a 25 |xm wire. This means that the centre of gravity of the avalanche is displaced towards the wire axis in the case of high gas gains; this effect is stronger if relatively thin wires are used. Besides the radial displacement of the centre of gravity a statistical spread in the f coordinate occurs, owing to the fact that the avalanche may spread non-symmetrically around the wire.

The displacement of the centre of gravity is confirmed by the measurements described in (6.1). We operated, therefore, with low gas gains in order to avoid the problems listed above.

When the avalanche is complete the electrons have reached the wire. The ions, having a drift velocity of an order of 1000 times lower than the electrons, are left behind close to the wire.

For each free electron under consideration in a proportional chamber the processes described above result in a tiny ion cloud which drifts from the wire to the cathode along the same field line along which the original electron approached the wire previously (fig. 1.3).

1.5 Applications

1.5.1 High Energy Physics

Since its invention in 1968 by Charpak et al. [14], the MWPC has had many derivatives, such as the drift chamber and the multistep avalanche chamber. Now they play a major role in all high-energy physics experiments. They are used in hadron and EM calorimeters and dE/dX counters, not only as track position measuring devices, but also as ionization chambers with gas gain.

Although the present thesis deals mainly with X-ray detection, our readout method could be applied to ionizing particles producing chamber-crossing tracks. The difference between detecting X-ray quanta and charged partirles lies in the distribution of the primary electrons. As indicated

22

above the X and Y-coordinates of the centre of gravity of the distribution are determined. In the case of ionizing particles the centre of gravity of the primary electron cloud is a good indicator of its position.

There is a consequence of the different shape of the charge distribution: in the case of soft X-ray quanta the ratio QUp / Q q0w n o f t h e tota' c a t n od e plane charges depends on angle <p (6.1). For charged particles crossing the chamber this ratio equals 1 (ignoring statistical fluctuations in electron-ion pair density along the track).

The readout system as described in this thesis is a processor capable of accepting a high counting rate in combination with good resolution. A drawback is the necessity of separate preamps connected to each wire or strip. In some cases it is possible to connect some strips together; two or more can be connected to one preamplifier. Off-line analysis can detect a 'hit' strip or wire by combining the signal distribution with data from elsewhere in the detector.

The spatial resolution in the case of tracks of minimum ionizing particles is affected by the emission of 9-rays. The Y-coordinate is particularly effected because of the relatively high probability of avalanches on two adjacent wires [4]. Such events can easily be suppressed (7.6).

The charge distribution over the strips is widened in the case of emission of a 3-ray. The change in the distribution width can be recognized and a veto issued.

A practical form for a high-accuracy track detector would be a chamber consisting of a number of layers of strip chambers. Criteria for accepting an event may cause one layer not to be fully efficiënt; the total detector, however, would still be fully efficiënt. This detector would combine high accuracy with high efficiency.

It should be realised that positional measurements of avalanches relate to the position of the wires and strips. Careful mechanical positioning of wires and/or foils with printed strips should therefore be considered at the design stage.

The readout circuit, discussed in the present thesis, consists of preamps, discriminators, a multiplexer and a signal processor. The number of signal processors required is equivalent to the number of tracks to be processed simultaneously.

Since the bulk price of charge-sensitive hybrid preamps is not very high, the total cost of the readout system is comparable to the price of commercially available drift chamber electronics. Inherent advantages of a strip chamber combined with the readout system are:

- the spatial resolution is not affected by the diffusion of electrons; - the influence of magnetic fields can be easily avoided;

- insensitivity to changes in gas temperature, pressure and mixture; - the use of a simple, low-cost, inexplosive gas mixture;

- the centre of gravity of the primary electron cloud has a close relation to the position of the partiele by which the cloud was created; and - the fact that it does not require an external trigger.

1.5.2 Nuclear Physics

Here also the use of wire chambers is widespread. The energy of charged particles emitted in a nuclear reaction is sometimes so low that it is important for the detector to be of low density. Wire chambers with very thin mylar gas foils (4 u.m) can be used [15]. Sometimes the high counting rate is a problem.

In nuclear physics photons with an energy from a few keV to about 10 MeV occur. With photons of 100 keV the efficiency of a gas filled wire chamber is very low. It can be improved by covering the chamber with a convertor. Much work has been done on the detection of 511 keV quanta from positron annihilation processes (drift tubes, drift multiiayers) [16]. The spatial resolution here is limited to about 2 mm. The electronics of a detector of this kind could be simplified by using wide strips, combined with the readout system of this thesis.

For the detection of neutrons another absorber should be used to convert an incident neutron into one or more charged particles by a nuclear reaction or a collision. It is also possible to use the gas of a MWPC itself as neutron convertor by filling it with a special gas such as 3H e .

1.5.3 MWPCs as a substitute for photographic film Medical applications

The detector could replace photographic film in X-ray diagnosis. This would have the advantage of obviating film handling; the data from the detector would be processed by computers and could be stored easily. Another improvement might be the theoretically unlimited dynamic range of the detector; whereas a film has a limited transmission ratio between the unexposed part and the blackest part, the dynamic range of a digital memory is virtually unlimited.

The efficiency of the detector drops severely at energies higher than 50 keV. It could be used in X-ray diagnoses for hands, arms, legs etc. since the quanta involved here are of about this energy level. For a picture through a complete body energies of up to 120 keV would be needed; here the detector would have to operate at extremely high pressure in order to ?A

reach 50 percent efficiency.

In miniature form the detector could well provide a substitute for the X-ray camera as commonly used by dentists.

X-ray diffraction

If a crystal is irradiated with monoenergetic X-rays a part of the quanta is reflected by the crystal planes at specific angles because of the wave appearence of the particles. A modern application of this phenomenon is the Guinier camera [17]. The radiation is absorbed by a photographic film whose transparency is measured later. This results in a spectrum which is characteristic of the crystal geometry of the sample. This device has been replaced by a chamber. The chamber has to be curved to prevent an unacceptable parallax error. Because of the curvature a wire could not be used as an anode; a bented razor blade was used instead. The induced charge of the avalanche ions was measured with the aid of strips and a delay line [17]. Another group used a strip readout method using a multi-ADC system [18]. For high counting rates, such as may occur sometimes in Synchroton Radiation Accelerators, the readout system described in the present thesis could be used.

Astrophysics

Modern astrophysics studies the whole spectrum of radiation from the universe. Since low-energy X-ray quanta are absorbed by the atmosphere, high altitude balloons, rockets and satellites have to be used to defect them.

A major problem in detecting X-rays with energies from 1-100 keV is to find their direction. This was once solved by placing an array of parallel pipes, with their axis in the Z direction, in front of a two-dimensional X-ray detector, allowing only perpendicular X-rays to be detected. This device measures only the radiation from a small region around its azimuth.

A recent development is the 'Coded Aperture Detector'; this consists of a two-dimensional X-ray detector with an absorbing mask in front of it. Suppose the mask had only one small hole: then the image of the X-ray sources observed in space would appear as a pinhole image on the detector. The efficiency of this device would be very low. The efficiency should be doubled if a second pinhole were made; numerical deconvolution of the image would produce one image again. More pinholes could be used: the ultimate is a mask which is transparent at known places. An example is shown in fig.1.16. Recently some groups have put great effort into finding the optimum 'pseudo-Gaussian' coded mask [19].

Chapter 2

Snduced eliarge ©o tSie electrodes

of a MWPC

2.1. Introduction

Charge signals on wires and strips in proportional chambers are caused mainly by a drifting cloud of positive ions [20,21,22]. The relations between these signals depend on the position of the cloud; in turn there is a relation between the position of an X-ray conversion point or track and the ion cloud. There is uncertainty as to the position and shape of the cloud owing to

- the finite path length of gas-ionizing electrons, - the finite range of fluorescent X-ray quanta, - diffusion of drifting electrons,

- statistical fluctuations in the avalanche mechanism, and

- diffusion of drifting ions due to collisions and mutual repulsion forces.

The ion cloud can be regarded as a superposition of point charges. Let us first ignore the effects mentioned above: we are then essentially concerned with the electrostatical problem of the induced charge on grounded conductors due to the presence of a point charge somewhere between them. The finite size of the charge cloud may be taken into account later, if necessary.

Given the chambers in combination with 55Fe radiation, the assumption of

a point-like space charge is adequate; see (1.2.2) and (5.4). The measurements of ion drift velocity (2.6-2.7) showed the effects of diffusion of the ion cloud; these can usually be ignored.

There are various ways of calculating induced charges:

a) Analytical methods using Green's Theorem: the charge distribution over the electrodes of some types of chambers can be calculated with the aid of simplifications such as mirror images and symmetries allowing replacement of point charges by line charges [20,23-32]. The approaches are not capable of calculating the charge distribution over a three dimensional chamber: in general only the distribution over the cathode planes is obtained.

b) Assume functions which describe the charge densities on the surfaces of the wires and cathode planes; calculate the total energy stored in this system of surface charges [33]. Vary the charge distributions in size and width until the minimum stored energy is obtained.

A problem is that one never knows when the real minimum energy is reached; it is hard, therefore, to estimate the accuracy of the method. Other objections against the algorithm of [33] are: 1) the authors assume the charge density on the wires to be independent of the 9 -coordinate; 2) the principle of obtaining the charge distributions on the cathode planes by superimposing induced charges due to the space charge and the induced charge on the wires is not valid since the total induced charge on the wires decreases with the charge that equals the induced charge on the cathode planes; 3) the assumption that the charge distributions over different wires can be described using the same function but with different parameters is not justified. c) Solution of the Laplace equation:

J E . d S = 0 {2.1} Integrating is done over a closed surface.

This equation is equivalent to:

A V = 0 {2.2}

where V is the potential of any point in the chamber gap. For a real three-dimensional solution the chamber gap has to be divided into small boxes. Since the size of the boxes depends on the geometry of thin wires either an extremely high number of boxes is needed or regions with different box sizes have to be defined. The latter is difficult: the algorithm depends on the chamber geometry. If we are interested only in the charge distribution over strips parallel to the wires and over the wires themselves the point charge may be assumed to be a line charge parallel to the wires, thus reducing the problem to a two-dimensional case.

d) A multi-particle simulation combined with a finite difference algorithm. This uses only Coulomb's Law and the fact that grounded conductors have zero potential at all points on their surface and internally. This method was developed to solve this electrostatic problem (2.2-2.3).

23

2.2 Reprint:

330

\ CALCULATION IN THREE DIMENSIONS OF

ELECTRODES OF AN MWPi

II. VAN DER G R A A F and J P. WAGENAAR Radiation Technologj Group, Physiu Drpafiment, Del/t Uminrsit)

I [fjtraduction

\n MWPC with cathode strips perpendicular to the wires can delermine botri coordinates of an X-raj

.is described elswhere [l|. Oplimisation ol this system requires a precise knowledge of the shape ,tnd magni­ tude "i" the signals on the wires and strips; this cannoi be calculated in an analytical waj and is nut eas> to measure In this paper we discuss a computer calcuta-lion and we present uur prelimmarj results aftei 6000 h PDPI i compuiing time.

2. The method

0167-5087 83/0000-0000, S03.ÜQ ! 1983 North-Holland

THE INDUCED CHARGE ON THE

,-J charge on ihc wires and cathode strips perpendicular i>> the uirtv

3 The program

_.\:

the toial n u m b e r of e l e c t r o n s involved. N o u we have a criierion For e l e c t r o d e A if an e l e c t r o n reaches ihe limit of .in e l e c t r o d e , then this e l e c t r o d e has too m a m elec­ t r o n s We can d e l e r m i n e e l e c t r o d e s with too Few elec­ t r o n s by c a l c u l a t i n g the force that would act on an e l e c t r o n placed on the limits of the e l e c t r o d e s ; since the p o l e n t i a l of each e l e c t r o d e is the sarne in case of e q u i -l i h n u m . ihe e -l c c -l r u d e which is re-lativdy the bhortes-l uf e l e c t r o n s will " p u l ] - the h a r d e s t on this edge (test)

X) /FR

After each tteration uil the e l e c t r o n s are tested on b e i n g out of limit If so. such an electron is placed ai r a n d o m on the surface of the most p u l l w g e l e c t r o d e

4 . hirsl test of the mi-lhod: hvo plano s j c o m e l n

j q u ü i b r i u m was the a b s e n c e ol significanl c h a n g e s :he d i s t r i b u t i o n of the i n d u c e d c h a r g e This resuli '■ ) b t a t n e d after 1500 i t c r a t i o n s t a k i n g 4K h C P U time i P D P 1 1 / 6 0 c o m p u t e r .

W e starled with a c a l c u l a t i o n of the i n d u c e d c h a r g e o n i w o parallel p l a n e s d u e lo a p o i n l c h a r g e c e n t e r e d in b e t w e e n ; since this p r o b l e m can bc solved in an a n a l y t i -cal way [5] it is a good test for the q u a l i t j of the a p p r o a c h . A p o i n l c h a r g e with value + 9 0 0 is placed in the origin the p l a n e s on Z = ± 5 m m , In fig. I the h i s t o g r a m s h o w s for a p l a n c the n u m b e r of e l e c t r o n s on 1 m m w i d e s t r i p s . T h e edges of both p l a n e s were set on A ' = ± 12.5 m m , Y = ± 12.5 mm, T h e h i s t o g r a m agrees very well with the theoreiical curve [6], T h e total n u m ­ b e r ol e l e c t r o n s on each p l a n e is 4 5 0 ± I so for this g e o m e t i y ihc i n d u c e d d i a i g c is e.ileul.ited wilh an accu-racy of 0 . 2 * . T h e criterion for having reached the

5, W i r e c h a m b e r gec

W e i i u v c o n s i d e i Ihe sa me e

firsi the elc and i

o n d u c i i n g pipes with i h m walls. T h e

diss i r a n diss l a t e d , like before, in the d i r e c -tion oi" the wirc-axis ( J f - d i r e c t i o n ] p r o p o r t i o n a l to Ihe X c o m p o n e n l of the force Then the e l e c t r o n is d i s p l a c e d hy a r o t a t i o n of ihe p o s i t i o n v e c t o r P (lig. 2). T h i s r o l a i i o n is p r o p o r t i o n a l with the cross p r o d u c t of ihe p o s i t i o n vector a n d the r e s u l t a n t of the Forces /■", a n d F

Fig. 3 s h o w s the g e o m e t r y of o u r c h a m b e r c a t h o d e p l a n e s a l Z = ± 5 m m , wires in the Z = ü p l a n e . g o i n g I h r o u g h Y = N > 2 m m ; N is the wire n u m b e r ; d i a m e t e r wires 20 firn T h e c e n t r a l wire 0 is the a v a l a n c h e wire;

Fig 3, Wire ehamber gcome disiancc R from ihe avolanche ■ an angle j> wilh the wire planc;

ry. The poinl charge i

\ W I R I - . ' ] ] \ M l i [ - R lJARAMI T k R S

30

t h e p o i n t c h a r g e is p l a c e d al X = 0, Y ^ R cos <f>, Z = R sin £; R is the d i s t a n c e of the wire axis.

rodei of on MII FC

Again we slarled with a r a n d o m d i s t r i b u t i o n of 91 e l e c t r o n s o v e r the c a t h o d e p l a n e s . T h e p o i n t c h a r g e w p l a c e d at R = 300 firn, <f> = 4_s" T h e limits of p l a n e s ar wires w e r e set at X= ±15 m m . Y= ± 7 . 5 m m . Aft

10000 iterations, taking one week CPU e q u i

-After this we varied R a n d $ s y s t e m a t i c a l l y ; for each n e w p o s i t i o n of the p o m t c h a r g e it t o o k a b o u t i w o d a y s C P U time to reach a new e q u i l i b r i u m .

f i g 4 s h o w s a p e r s p c c t i v e view of the e l e c t r o n d i s t r i b u t i o n ; R = 1 0 0 / m i , $ = 2 0 ° . Fig 5 s h o w s a detail of ihe a v a l a n c h e wire; the p o s i t i o n of ihe p o i n l c h a r g e is m a r k e d wilh a c r o s s . Fig 6 s h o w s the lotal c h a r g e o n t h e a v a l a n c h e wire as a funciion of R, we n o t i c e d that this c h a r g e is a l m o s l i n d e p e n d e n t of £ . T h i s c h a r g e is Ihe s u m of Ihe e l e c t r o n c h a r g e that d n l ' t e d Lo the wire d u n n g the a v a l a n c h e . a n d the p o s ï t i v e i n d u c e d c h a r g e d u e to ihe p o i n t c h a r g e In o u r case ihe firsl c h a r g e e q u a l s - 9 0 0 .

Fig. 7 is a h i s t o g r a m similar lo fig 2 for the wire c h a m b e r g e o m e t r y . W e n o t i c e d ihat for R < 3 0 0 u r n ihe s h a p e of t h e d i s t r i b u t i o n over ihe strips is n o l very d e p e n d e n l o n R a n d <£ In spiie of the p o o r statistics o n e m a y c o n c l u d e ihat the d i s t r i b u t i o n looks like a G a u s s i a n c u r v e , a n d that its w i d t h is a b o u i 20% s m a l l e r t h a n ihe t w o p l a n e g e o m e t r y case. T h i s is n o l in agreee-m e n i with agreee-m e a s u r e agreee-m e n t s d o n e b \ E n d o et al. | 6 | .

In fig. 8 the r a t i o of the total i n d u c e d c h a r g e Q U of the u p p e r p l a n e a n d Q D of the d o w n p l a n e is p l o t l e d as a funciion of <j>. I n fig. 9 the r a t i o of the i n d u c e d c h a r g e Y Q R on wire 1 n g h l of the a v a l a n c h e wire a n d ihe toial c h a r g e Y Q M of the a v a l a n c h e wire is p l o t l e d as a

nl charge

v of the electrons i

n the calhode plani

equihhnum Bolh

Fig 8 The ratio of lh cathodc plane as a turn

R = 100 ura

RNGLE (2 10EG.] charge on the upper

n o f £ T h e r a t i o of t h e c h a r g e s found lo c a r r y n o o b s e r v a b l e i

In fig. 10 the p l o t l e d as J funct.

7. Signal d e v e l o p m e n l

S u p p o s e we c o n n e c t the wires a n d strips via capac t o r s 10 g r o u n d . T h e voltage o v e r the c a p a c i t o r s will 1 p r o p o r l i o n a l to the c h a r g e s c a k u l a t e d a b o v e ; the devi

. R = 100 um

> R = 150 pm

e s o f Y Q R VQM a s a lunet ion of the

o p m e n t of these signals is a funclion of the p o s i t i o n of

the ion cloud The space-time relation of the ion cloud

e a n b e o b l a i n e d in several ways: o n e c a n m e a s u r e i h e v o l t a g e - t i m e r e l a t i o n of the a v a l a n c h e w i r e . c o m b i n e the results with the s i g n a l - s p a c e r e l a t i o n of fig r> A n o i h e r m e t h o d i.s t h e h o m o g e n e o u s i r r a d i a t i o n af t h e e h a m b e r with X - q u a n t a ; m e a s u r e the e x t r e m e values of Y Q R / Y Q M as a ï u n c n o n erf time 1t = 0 signal s t a r t s ) T h e ratio Y Q R / Y Q M is the highest for ^ - 0 ° , a n d the lowesl for 0 = 180°. T h e s p a i c - t i m c relation can he read from fig 10, Finally o n e ean d i r e c t l y m e a s u r e or c a l e u l a t e t h e drift velocily of ïens in gases as a function of t h e eleclric field. If the signals a r e p r o c e s s e d b> a p r e a m p l i f i e r . t h e n t h e influence of a fimte b a n d w i d l h h a s l o b e l a k e n in a c c o u n t S m c e the s h a p e of the d i s t r i h u t i o n over the strips does nol d e p e n d v e n ' m u c h o n R a n d r> we m a y c o n c l u d e t h a l a limiied b a n d w i d l h d o e s n o t p l a y an i m p o r t a n t role in the signal d i s t r i b u -tion from the s t r i p s . Signals from ihe W'ires. h o w e v e r , d o h a v e differences in rise t i m e , as c a n be e o n c l u d e d from d g . 10.

T h e e r r o r , c a u s e d by the fact ihat the c e n t e r of g r a v i t y of the ion c l o u d starts a few m i c r o n s off t h e a v a l a n c h e wire surface may be neglected, as c a n be d e n v e d from fig 6.

R.UGLE 0 I 0 E G . I

Fig. 9 The ralio of ihe nghl wire charge and the a charge as a (uncLion of £

9. F u t u r e vvork

W e have the possibility of u s i n g four h o u r s C P U t i m e of the C y b e r 205 S u p e r c o m p u t e r in M i n n e a p o h s . U S A . In o r d e r to e o m e lo a c o m p l e t e d e s c r i p t i o n of ihe

X WIRE CHAMBER PARAMEÏFKS

32

We would like to ttumk ihe Rdumr Physics Group, speciallj Ir r' f.A. d^ Leege for offering their PDIJ

2.3 P r e p r i n t

Numerical results of calculations in three

dimensions of the induced charge in MWPCs

Harry van der Graaf

NIKHEF-H, Kruislaan 409, Amsterdam, The Netherlands

and

Jan Pieter Wagenaar

Radiation Technology Group, Delft University of Technology, The Netherlands

Present adress: Hollandse Signaal Apparaten, Hengelo, The Netherlands Submitted to Nucl. Instr. and Methods (Feb. 1986).

An algorithm has been updated and vectorized for the calculation of induced charge distributions in proportional wire chambers. On a Cyber 205 Supercomputer the algorithm is fast and accurate. Results are in agreement with existing approaches and

measurements. New details of the cathode charge distributions and wire signals as a function of the point charge position are presented.

1. Introduction

The muiti partiele approach and some results have been presented before [1]. The charge signals from the conductors of a MWPC can be represented by the charge that flows from ground into the grounded conductors if a positive point-shaped ion charge is placed somewhere near the avalanche wire. The avalanche wire charge signal is the sum of the negative electron charge created in the avalanche and the charge induced by the ion cloud, which appears as a positive charge.

in the situation of an electrostatic equilibrium no tangential force acts on the electrons at the conductors. The potential is zero on the surfaces and in all conductors.

Some analytical approaches exist [2-5]. The multi-particle method, however, is a general way of calculating three-dimensional electrostatic charge distributions and can be applied to exotic chambers such as the blade chamber [6] or needie chambers [7]. It will be shown that accuracy depends only on the CPU time available.

The algorithm has been vectorized for the Cyber 205 Supercomputer. The CPU time needed to calculate the distribution for a particular chamber geometry is reduced to one hour if this machine is used

2. The algorithm

As explained in the previous paper [1], we start with a random distribution of N electrons over the surfaces of wires and cathode planes of a MWPC. The electrons have a charge of - 1 ; a positive point charge of + N is placed somewhere in the chamber gap. Three coordinates of each electron and the point charge are stored in an array. Now the total force vector on each electron due to the presence of the other electrons and the point charge is calculated according to Coulomb's Law.

After all the forces have been computed the electrons are displaced a step in the direction of the resulting force, with the restriction that they may not leave the surface of the conductor. New positional coordinates replace the old ones; the forces are all computed again, resulting in the next displacement. This iterating process continues.

Whereas in the previous paper criteria for electron transitions between

",!,

conductors were based on calcuiations of force on charges placed at the outer ecges of the chamber. here we apply a enterion using equal potentials. II was found that the edge definition caused a systematic error which could only be reduced sufficiently at the expense of an unacceptable CPU time.

We calculate the potential at point i as: vi - £ Qj/Rij

J

where FL is the distance frorn point i to electron or point charge Q;: alt etectrons and the positive point charge contribute. The potential of a conductor is defined as the average of the potentials at many points in or on that conductor. A point where a potential is calculated acts as a 'probe'. In the case of wires there were 51 probes on the axis of the wire; on each cathode plane we defined a probe grid of 561 points.

Too many electrons on a conductor make its potential low. After each electron displacement procedure the potentials of cathode planes and wires are calculated; the conductors with the minimum and maximum potential respectively are selected. These exchange an electron. In order to disturb the distributions as little as possibte the most off-centre electron on the low-potential conductor is selected to change electrode. This electron is positioned on the high-potential conductor at coordinates ( - Xm a x i -Ypnax)' xrnax a n c l Yrnax D e l n9 t h e coordinates of the most off-centre electron en the high-potential conductor. See fig.1 for the coordinate system.

In this way we avoided the edge limit needed for the force criterion. If, however, an electron happens to be very close to a potential probe, the calculated potential will be too low. After one iteration this will not be the case. We therefore expect 'noise' and negative spikes in the potentials. An acceptable noise level requires a minimum number of potential probes. Optimization of the algorithm

Most CPU time was spend on the force routine. A vectorized subroutine was developed and optimized, discussed in detail in [8], The algorithm is ideal for this kind of 'long-pipeline' vector processor; a speed gain factor of 110 versus Amdahl V7B and 1150 versus PDP 11/60 was obtained. Other improvements were obtained by optimizing the electron stepping following each force calculation. In principle the step is proportional to the tangential force component and with a 'step' constant chosen empiricaily. A low step constant implies a long CPU time before equilibrium is achieved; a high one results in electron 'overshoots'. If

systematic differences occur. Fig.2 shows the electron distribution for the Standard chamber with N = 4000.

Fig.2 Bird's-eye view of the electron distribution in the Standard chamber; number of

participating electrons N is 4000. The 'clouds' above and beneath the wire pfane are formed by the electrons in the calhode planes. The point charge (+4000) is marked with a cross. Note the regular patterns in the cathode electron positions.

(1), (2) and (3) It takes about 200 iterations to move from one equilibrium to another, therefore 500 iterations were done each time, foliowed by 500 more for a better average. The accuracy for the charge on a conductor was 2 electrons in the case of a 1000 electron avalanche. Fig.3 shows potential and charge changes of wires -1 and 1 if the ions in the Standard chamber jump from 9 = 90 to 9 =0° and back.

(4) Distributions were reproduced; no dependency on previous situations was observed.

(5) A calculation with 4000 electrons resulted in the same distributions as with 1000 electrons. Using 4000 electrons could result in higher accuracy but at the cost of much more CPU time, since the CPU time per iteration is proportional to the square of the number of electrons. The same CPU time can also used for more iterations in equilibrium, resulting in a more accurate average value. The optimum value for N is not yet

3? 8 4 < 0 ü -4 O Q-~80 40 80 V2Ö f ê ö 2 0 0 2 4 0 28Ö" 1 2 0 | 'TERATION 100 80 ë 60 _< x <-> 4 0 i 1 1 1 1 1 . 1 , 0 40 80 120 160 2 0 0 2 4 0 2 8 0 ITERATION

Fig.3 Potentiai and charge changes of wire 1 and 1. The system was in equilibrium: f

-90°. At iteration 0 this angle was changed to 0°. The potentials change immediately, but since an average is taken of the last 10 values the change is smcothed out. Because of the potential change electrons start to flow from wire -1 to wire 1. At iteration 100 the point charge is set back to<p = 90°. and back toiP at iteration 200. Note the noise in the potentials and the charges. The wire charges are expressed in electrons; N ~ 1000.

known.

Using a limited number of electrons seems to introducé a systematic error, as shown in fig.4, which plots the potential on the avalanche wire axis as a function of X. In the ideal case these potentials are all zero. In the case of a limited number of participating electrons some spread and a few negative 'spikes' can be expecfed. However, the potential is systematically too negative in the region of the ion charge (X = 0) and positive on either side, ending in slightly positive tails. This can be explained by the appearance of regular patterns at regions of constant surface charge density ( for all wires and planes this is near X = 0). At a certain distance from the centre of the chamber the pattern is broken; there the electrons move all the time, causing greater 'pressure'. This results in a local charge density which is too low. The disturbed distributions over wires and planes are therefore too narrow.

The effect, however, is very small. From fig.4 it can be seen that the redundant charge at X = 0 for the avalanche wire is only 6 and 3 percent in the 1000 and 4000 electron cases respectively. This surplus charge is compensated for in the nearby regions; the total charge on the wires is not affected. The same effect is very small on the cathode planes. We may assume that these distributions are not affected. The K3 parameters

39

- - - - W I R E - 1 WIRE + 1

4 0 0 < E; 2 0 0 9 n D. 0 - 2 0 0 - 4 0 0 - 6 0 0 - 8 0 0 - 1 0 0 0

\fi

w

Fig.4 Potentia/ at the avalanche wire axis as a function of X. Note the negative spike in the N

= 1000 electron curve atX = 3 mm.

describing the cathode distributions were the same in the case of 1000 and 4000 electrons.

5. Charges on wires

Fig.5 plots the total avalanche wire charge ( = avalanche charge + induced charge) against R, S and D. This charge Qa v is expressed as a fraction of the total avalanche size N. A time-dependent function for Qa v can be obtained by combining the ion mobility equation w = u. E (see [10]) with Erskine's expression for the electrical field [9]. The log dependence of Qa v on R agrees well with Mathieson's expression [3]; the equivalent coaxial cathode radius is 4 mm.

Fig.6 shows that Qa v does not depend on >p in the region R = 0.05 - 0.8 mm. This means that with R < 0.8 mm the avalanche wire signal can be described as a pulse from a proportional tube with equivalent cathode radius Rc. This radius may be calculated for a particular wire chamber geometry [3].

Fig.7 shows the ratio Q-J/QQ as a function of <f> for the Standard chamber. Because of the symmetry in the chamber geometry the ratio Q.-j/Qg can be

40

-0,/N (S)

»tVN (D)

t-ig.5 S , R . 0 1 mrr. \U)

100 /im (Dj Total avalanche wire charge as a function of wire spacing S, point charge radia! distance R and wire diameter D. The charge is expressed as fraction of the total a valanche size.

O (DEG)

Flg. 7 The charge on wire 1 as a function of f. The curve is a least square fit.

R=0.8

-'

Avalanche wire charge as a function of f for some values of R, expressed as fraction of the total avalanche size.

considered as:

Q_l/Q0(f> ) = Q1/ Q0( * + 1 8 0 ° ) We fitted the data using the expression

Q-|/Q0(<P ) = A0 + A1c o s ( f ) + A2cos(2>p )

This Fourier series contains only cosine terms since the first derivative has to be zero for f = 0° and f = 180°.

.^■1 w

I . ; ttTOCh p t e ■

■

. . . .

o □ .34 0.32 0.3 0.28 0.26 175 200 225 250 21'. » (OEG) Fig.12

The total charge on the upper cathode Qup as a function of f. The curve is a least square fit

Fig.13 CQ C-J andC2asa function of R.

Fig.14

C()C1and C2as a function ofS. Note that with infinite S the constant C0

is 0.5.

It fe interesting to note that the ratio Qu p/Q0 also does not change with R

and is constant in time.

4 4

Measurements of charges on wires and cathodes

Fig.15 plots the ratio QUp/Qd0wn a9a i n s t Q-|/Q-1 f o r e a c n event of

absorption of an 5 5Fe quantum by the chamber. Similar results were

obtained first by Charpak et al. [11]. It shows an ellipse, elongated in two directions. The graph may be considered as an image of avalanches arriving at the wire. The extreme values (A0+ A.|)/(A0 - A-|) and (C0 + C-|)/(C0 - Cj)

can be reconstructed as 1.3 and 1.2 respectively, indicating a distance of 200 urn between the ion cloud and the avalanche wire axis.

This tallies with the calculated values, assuming an integration time of the charge-sensitive ADC of 300 ns and an ion mobility constant n = 0.17 um2/V.ns [10].

Fig. 15

The measured ratios Qur/Ocjown

plotted against Q.j/Qp Note the deformed ellipse due to non-linearities too far away from (i, 1). The preamps were adjusted so that the vertical line X = 1 was a tangent to the ellipse.

1 2

Ol/Q-1

7. Cathode plane charge distributions

Fig.16 shows the Standard chamber charge distribution in the X direction perpendicular to the wires: the charge is integrated here in the Y direction. Because f equals zero, both X distributions of upper and lower cathode plane are the same; the Y distributions are also the same. Fig.17 shows the Y distribution. In the tails the charge density of the Y distribution decreases faster than in the X distribution.

All cathode distributions were parameterized using the Gatty-Mathieson single parameter expression for normalized distribution [5]:

c

l

O

O

'—

ïoi^y.''

w = K

Fig.16 Charge dislributton over strips perpendicular to the wires. Average of DX and UX. Dashed curve: Gatty-Mahieson single parameter approach: Kg = 0.36

In the case of Y distributions with f different from 9 0 ° we took inio account the shift in the centre of gravity of the distributions, referred to as AYD and AYU for the lower and upper cathode plane:

x = (Y-AY)/H

Fig.18 shows K3 for the DX, UX, DY and UY distributions as a function of S. This is in good agreement with values found by Mathieson et al [3], It is obvious that where S > H the X and Y distributions are the same, causing a circular symmetrical charge distribution on the cathodes. With narrow wire spacings this is not the case.

Fig.19 plots the K3 values for <p = 90°. A striking difference between the upper and lower plane appears: both X and Y distributions are narrower at the top cathode. Consequently the charge distributions depend on the incident angle of approaching drifting electrons.

■*2£L

Fig.17

Charge distribution over strips

paratlei to the wires. Ko = 0.16; AY = 0.30.

46

Fig.18

Vaiue K3 pioited tor the upper cathode plane distnbutions UX and UY and lower cathode distributions DX and DY as a function otS.

Fig.19 As fIg. 18; f - 90' 0 ■ \ ^ r • DX ■ UX * DY ■ UY \ l

'

should be identical; also DY and UY.

The DX distribution may suggest a minimum for K3 at S = 1 mm. Since the total number of electrons on the cathodes at wire spacings smaller than 1 mm is fairly small the K3 figures in this region are less reliable. Repeated calculations, however, showed the same effect.

Fig.20 shows the dependency of K3 on R. The cathode charge distributions 47