Signal Processing for RadiationDetectors

Signal Processing for Radiation Detectors

Mohammad Nakhostin

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Library of Congress Cataloguing-in-Publication Data

Names: Nakhostin, Mohammad, 1973– author.Title: Signal processing for radiation detectors / by Mohammad Nakhostin.Description: Hoboken, NJ, USA : Wiley, 2018. | Includes bibliographicalreferences and index. |

Identifiers: LCCN 2017023307 (print) | LCCN 2017024048 (ebook) | ISBN9781119410157 (pdf) | ISBN 9781119410164 (epub) | ISBN 9781119410140(hardback)

Subjects: LCSH: Nuclear counters. | Radioactivity–Measurement. | Signal processing.Classification: LCC QC787.C6 (ebook) | LCC QC787.C6 N35 2017 (print) | DDC539.7/7–dc23

LC record available at https://lccn.loc.gov/2017023307

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To my daughter, Niki

Have patience. All things are difficult before they become easy.Saadi, Persian poet

Contents

Preface xiAcknowledgement xiii

1 Signal Generation in Radiation Detectors 11.1 Detector Types 11.2 Signal Induction Mechanism 21.3 Pulses from Ionization Detectors 91.4 Scintillation Detectors 57

References 72

2 Signals, Systems, Noise, and Interferences 772.1 Pulse Signals: Definitions 772.2 Operational Amplifiers and Feedback 802.3 Linear Signal Processing Systems 832.4 Noise and Interference 1012.5 Signal Transmission 1202.6 Logic Circuits 130

References 133

3 Preamplifiers 1353.1 Background 1353.2 Charge-Sensitive Preamplifiers 1373.3 Current-Sensitive Preamplifiers 1593.4 Voltage-Sensitive Preamplifiers 1623.5 Noise in Preamplifier Systems 1633.6 ASIC Preamplifiers 1763.7 Preamplifiers for Scintillation Detectors 1823.8 Detector Bias Supplies 186

References 187

vii

4 Energy Measurement 1914.1 Generals 1914.2 Amplitude Fluctuations 1944.3 Amplifier/Shaper 2034.4 Pulse Amplitude Analysis 2344.5 Dead Time 2444.6 ASIC Pulse Processing Systems 249

References 256

5 Pulse Counting and Current Measurements 2615.1 Background 2615.2 Pulse Counting Systems 2635.3 Current Mode Operation 2745.4 ASIC Systems for Radiation Intensity Measurement 2865.5 Campbell’s Mode Operation 289

References 293

6 Timing Measurements 2956.1 Introduction 2956.2 Time Pick-Off Techniques 3006.3 Time Interval Measuring Devices 3206.4 Timing Performance of Different Detectors 330

References 345

7 Position Sensing 3497.1 Position Readout Concepts 3497.2 Individual Readout 3537.3 Charge Division Methods 3577.4 Risetime Method 3737.5 Delay-Line Method 375

References 380

8 Pulse-Shape Discrimination 3838.1 Principles of Pulse-Shape Discrimination 3838.2 Amplitude-Based Methods 3868.3 Zero-Crossing Method 3938.4 Risetime Measurement Method 3998.5 Comparison of Pulse-Shape Discrimination Methods 401

References 404

9 Introduction to Digital Signals and Systems 4079.1 Background 4079.2 Digital Signals 408

Contentsviii

9.3 ADCs 4149.4 Digital Signal Processing 418

References 438

10 Digital Radiation Measurement Systems 44110.1 Digital Systems 44110.2 Energy Spectroscopy Applications 44810.3 Pulse Timing Applications 47210.4 Digital Pulse-Shape Discrimination 483

References 498

Index 503

Contents ix

Preface

Ionizing radiation is widely used in various applications in our modern life,including but not limited to medical and biomedical imaging, nuclear powerindustry, environmental monitoring, industrial process control, nuclearsafeguard, homeland security, oil and gas exploration, space research, materialsscience research, and nuclear and particle physics research. Radiation detectorsare essential in such systems by producing output electric signals wheneverradiation interacts with the detectors. The output signals carry informationon the incident radiation and, thus, must be properly processed to extractthe information of interest. This requires a good knowledge of the characteris-tics of radiation detectors’ output signals and their processing techniques. Thisbook aims to address this need by (i) providing a comprehensive description ofoutput signals from various types of radiation detectors, (ii) giving an overviewof the basic electronics concepts required to understand pulse processingtechniques (iii), focusing on the fundamental concepts without getting toomuch in technical details, and (iv) covering a wide range of applications so thatreaders from different disciplines can benefit from it. The book is useful forresearchers, engineers, and graduate students working in disciplines such asnuclear engineering and physics, environmental and biomedical engineering,medical physics, and radiological science, and it can be also used in a courseto educate students on signal processing aspects of radiation detection systems.

Mohammad NakhostinGuildford, Surrey, UKJuly 2017

xi

Acknowledgement

I owe a special debt of gratitude to some individuals who have had a strong per-sonal influence on my understanding of the topics covered in this book. Theyinclude Profs. M. Baba and K. Ishii and Drs. K. Hitomi, M. Hagiwara, T. Oishi,Y. Kikuchi, M.Matsuyama, and T. Sanami from the Tohoku University in Japan.I would like to extend my appreciation to the people at the University of Surrey,United Kingdom, for hosting me during the last couple of years. I should alsothank Dr. A.Merati for his help with the preparation of the text. I am thankful toBrett Kurzman, Victoria Bradshaw, Kshitija Iyer, and Viniprammia Premkumarat John Wiley & Sons, in the United States and India for handling this projectand for their advice on shaping the book. Finally, I gratefully acknowledgethat the completion of this book could not have been accomplished withoutthe support and patience of my spouse, Maryam.

xiii

1

Signal Generation in Radiation Detectors

Understanding pulse formation mechanisms in radiation detectors is necessaryfor the design and optimization of pulse processing systems that aim to extractdifferent information such as energy, timing, position, or the type of incidentparticles from detector pulses. In this chapter, after a brief introduction onthe different types of radiation detectors, the pulse formation mechanisms inthe most common types of radiation detectors are reviewed, and the character-istics of detectors’ pulses are discussed.

1.1 Detector Types

A radiation detector is a device used to detect radiation such as those producedby nuclear decay, cosmic radiation, or reactions in a particle accelerator. In addi-tion to detecting the presence of radiation, modern detectors are also used tomeasure other attributes such as the energy spectrum, the relative timingbetween events, and the position of radiation interaction with the detector.In general, there are two types of radiation detectors: passive and active detec-tors. Passive detectors do not require an external source of energy and accumu-late information on incident particles over the entire course of their exposure.Examples of passive radiation detectors are thermoluminescent and nucleartrack detectors. Active detectors require an external energy source and produceoutput signals that can be used to extract information about radiation in realtime. Among active detectors, gaseous, semiconductor, and scintillation detec-tors are the most widely used detectors in applications ranging from industrialand medical imaging to nuclear physics research. These detectors deliver attheir output an electric signal as a short current pulse whenever ionizing radi-ation interacts with their sensitive region. There are generally two differentmodes of measuring the output signals of active detectors: current mode andpulse mode. In the current mode operation, one only simply measures the totaloutput electrical current from the detector and ignores the pulse nature of the

1

Signal Processing for Radiation Detectors, First Edition. Mohammad Nakhostin.© 2018 John Wiley & Sons, Inc. Published 2018 by John Wiley & Sons, Inc.

signal. This is simple but does not allow advantage to be taken of the timing andamplitude information that is present in the signal. In the pulse mode operation,one observes and counts the individual pulses generated by the particles. Thepulse mode operation always gives superior performance in terms of theamount of information that can be extracted from the pulses but cannot be usedif the rate of events is too large. Most of this book deals with the operation ofdetectors in pulse mode though the operation of detectors in current mode isalso discussed in Chapter 5. The principle of pulse generation in gaseous andsemiconductor detectors, sometimes known as ionization detectors, is quitesimilar and is based on the induction of electric current pulses on the detectors’electrodes. The pulse formation mechanism in scintillation detectors involvesthe entirely different physical process of producing light in the detector. Thelight is then converted to an electric current pulse by using a photodetector.In the next sections, we discuss the operation of ionization detectors followedby a review of pulse generation in scintillation detectors and different types ofphotodetectors.

1.2 Signal Induction Mechanism

1.2.1 Principles

In gaseous and semiconductor detectors, an interaction of radiation with thedetector’s sensitive volume produces free charge carriers. In a gaseous detector,the charge carriers are electrons and positive ions, while in the semiconductordetectors electrons and holes are produced as result of radiation interactionwith the detection medium. In such detectors, an electric field is maintainedin the detection medium bymeans of an external power supply. Under the influ-ence of the external electric field, the charge carriers move toward the electro-des, electrons toward the anode(s), and holes or positive ions toward thecathode(s). The drift of charge carriers leads to the induction of an electric pulseon the electrodes, which can be then read out by a proper electronics system forfurther processing. To understand the physics of pulse induction, first considera charge q near a single conductor as shown in Figure 1.1. The electric force ofthe charge causes a separation of the free internal charges in the conductor,which results in a charge distribution of opposite sign on the surface of the con-ductor. The geometrical distribution of the induced surface charge depends onthe position of the external charge q with respect to the conductor. When thecharge moves, the geometry of charge conductor changes, and therefore, thedistribution of the induced charge varies, but the total induced charge remainsequal to the external charge q. We now consider a gaseous or semiconductordetector with a simple electrode geometry including two conductors as shownin Figure 1.2. If an external charge q is placed at distance x∘ from one electrode,

1 Signal Generation in Radiation Detectors2

Conductor

External charge q

YX

Surface charge density

Figure 1.1 The induction of charge on a conductor by an external positive charge q (top) andthe density of the induced surface charge on the conductor (bottom).

q1

q1

q2

q2

q

q

d i

Surface charge density

xº

Figure 1.2 The induction of current by a moving charge between two electrodes. Whencharge q is close to the upper electrode, the electrode receives larger induced charge, but asthe chargemoves toward to the bottom electrode, more charge is induced on that electrode.If the two electrodes are connected to form a closed circuit, the variations in the inducedcharges can be measured as a current.

1.2 Signal Induction Mechanism 3

charges of opposite sign with the external charge are induced on each electrodewhose amount and distribution depends on the distance of the external chargefrom the electrode [1]:

q1 = −qx∘d

1 1

and

q2 = −q 1−x∘d

1 2

where d is the distance between the two electrodes. When the external chargemoves between the electrodes, the induced charge on each electrode varies,but the sum of induced charges remains always equal to the external chargeq = q1 + q2. If two electrodes are connected to form a closed circuit, the changesin the amount of induced charges on the electrodes lead to ameasurable currentbetween the electrodes. As it is illustrated in Figure 1.2, when the externalcharge is initially close to the upper electrode, most of the field strength willterminate there and the induced charge will be correspondingly higher, butas the charge moves toward the lower electrode, the charge induced on thelower electrode increases. This means that the polarity of outgoing charges fromelectrodes or the observed pulses are opposite. In general, the polarity of theinduced current depends on the polarity of the moving charge and also thedirection of its movement in respect to the electrode. As a rule, one can remem-ber that a positive charge moving toward an electrode generates an inducedpositive signal; if it moves away, the signal is negative and similarly for negativecharge with opposite signs. In a radiation detector, a radiation interactionproduces free charge carriers of both negative and positive signs. The motionof positive and negative charge carriers toward their respective electrodesincreases their surface charges, the cathode towardmore negative and the anodetoward more positive, but by moving the charge away from the other electrode,the charge of opposite polarity is induced on that electrode. The total inducedcharge on each electrode is due to the contributions from both types of chargecarriers, which are added together due to the opposite direction and oppositesign of the charges.The start of a detector output pulse, in most of the situations, is the moment

that radiation interacts with the detector because the charge carriers immedi-ately start moving due to the presence of an external electric field. The pulseinduction continues until all the charges reach the electrodes and get neutra-lized. Therefore, the duration of the current pulse is given by the time requiredfor all the charge carriers to reach the electrodes. This time is called the chargecollection time and is a function of charge carriers’ drift velocity, the initial loca-tion of charge carriers, and also the detector’s size. The charge collection timecan vary from a few nanoseconds to some tens of microseconds depending on

1 Signal Generation in Radiation Detectors4

the type of the detector. By integrating the current pulse generated in the detec-tor, a net amount of charge is produced, which would be equal to the totalreleased charge inside the detector if all the charge carriers are collected bythe electrodes. In most of the detectors, there is a unique relationship betweenthe energy deposited by the radiation and the amount of charge released in thedetector, and therefore, the deposited energy can be obtained from the integra-tion of the output current pulse. Figure 1.3 shows the induced pulses when adetector’s electrode is segmented. The amplitude of the pulse induced on eachsegment will depend upon the position of the charge with respect to the seg-ment. As the charge gets closer to the electrode, the charge distributionbecomes more peaked, concentrating on fewer segments. Therefore, with aproper segmentation of the electrode, one can obtain information on the loca-tion of radiation interaction in the detector by analyzing the induced signals onthe electrode’s segments. This is called position sensing and such detectors arecalled position sensitive. Detectors with electrodes divided to pixels or strips arethe most common types of designs for position sensing in radiation imagingapplications. It should be also mentioned that the induction of signal on a con-ductor is not limited to the electrodes that maintain the electric field in thedetector. In fact, any conductor, even without connection to the power supply,can receive an induced signal. This property is sometimes used to acquire extrainformation on the position of incident particles on the detectors.The induced charge on an electrode by amoving charge q can be computed by

using the electrostatic laws. This approach is illustrated in Figure 1.4 where thecharge q is shown in front of an electrode. The induced charge Q on the elec-trode can be calculated by using Gauss’s law. Gauss’s law says that the inducedcharge on an electrode is given by integrating the normal component of the

q

q

Segmented electrode

Figure 1.3 The induction of pulses on the segments of an electrode. In a segmentedelectrode, charge is initially induced on many segments, but as the charge approaches theelectrode, the largest signal is received by the segment, which has the minimum distancewith the charge.

1.2 Signal Induction Mechanism 5

electric field E over the Gaussian surface S that surrounds the surface of theelectrode:

SεE dS =Q 1 3

where ε is the dielectric constant of the medium. The time-dependent outputsignal of the detector can be obtained by calculating the induced charge Q onthe electrode as a function of the instantaneous position of the moving chargesbetween the electrodes of the detector. However, this calculation process is verytedious because one needs to calculate a large number of electric fields corre-sponding to different locations of charges along their trajectory to obtain goodprecision. A more convenient method for the calculation and description of theinduced pulses is to use the Shockley–Ramo theorem. The method is describedin the next section, and its application to some of the common types of gaseousand semiconductor detectors are shown in Sections 1.3.1 and 1.3.2.

1.2.2 The Shockley–Ramo Theorem

Shockley and Ramo separately developed a method for calculating the chargeinduced on an electrode in vacuum tubes [2, 3], which was then used for theexplanation of pulse formation in radiation detectors. Since then, several exten-sions of the theorem have been also developed, and it was proved that thetheorem is valid under the presence of space charge in detectors. The proofand some recent reviews of the Shockley–Ramo theorem can be found in Refs.[4–6]. In brief, the Shockley–Ramo theorem states that the instantaneouscurrent induced on a given electrode by a moving charge q is given by

i= −qv E∘ x 1 4

and the total charge induced on the electrode when the charge q drifts fromlocation xi to location xf is given by

Q= −q φ∘ xf −φ∘ xi 1 5

q

EGaussian surface (S)

Figure 1.4 The calculation of induced charge on an electrode by using Gauss’s law.

1 Signal Generation in Radiation Detectors6

In the previous relations, v is the instantaneous velocity of charge q and φ∘and E∘ are, respectively, called the weighting potential and the weighting field.The weighting field and the weighting potential are a measure of electrostaticcoupling between the moving charge and the sensing electrode and are theelectric field and potential that would exist at q’s instantaneous positionx under the following circumstances: the selected electrode is set at unitpotential, all other electrodes are at zero potential, and all external chargesare removed. One should note that the actual electric field in the detectoris not directly present in Eq. 1.4, but it is indirectly present because the chargedrift velocity is normally a function of the actual electric field inside thedetector. In the application of the Shockley–Ramo theorem to radiationdetectors, the magnetic field effects of the moving charge carriers areneglected because the drift velocity of the moving charge carriers is low com-pared with the velocity of light. For example, in germanium the speed of lightis 750 × 107 cm/s, while the drift velocity of electrons and holes is less or com-parable with 107 cm/s. The calculation of weighting fields and potentials insimple geometries such as planar and cylindrical electrodes can be analyti-cally done, which enables one to conveniently compute the time-dependentinduced pulses. In the case of more complex geometries such as segmentedelectrodes with strips or pixel structure, one can use electrostatic field calcu-lation methods that are now available as software packages. In the followingsections, we will use the concept of weighting fields and potentials for calcu-lating the output pulses for some of the common types of gaseous and sem-iconductor detectors, but before that we describe how a detector appears assource of signal in a detector circuit.

1.2.3 Detector as a Signal Generator

We have so far discussed that ionization detectors produce a current pulse inresponse to an interaction with the detector. Therefore, detectors can be con-sidered as a current source in the circuit. Figure 1.5 shows the basic elements ofa detector circuit together with its equivalent circuit. The detector exhibits acapacitance (Cd) in the circuit to which one can add the sum of other capaci-tances in the circuit including the capacitance of the connection between thedetector and measuring circuit and stray capacitances present in the circuit.The detector also has a resistance shown by Rd. The bias voltage is normallyapplied through a load resistor (RL), which in the equivalent circuit lies in par-allel with the resistor of the detector. In a similar way, the measuring circuit,which is normally a preamplifier, has an effective input resistance, Ra, andcapacitance, Ca. When the detector is connected to the measuring circuit,the equivalent input resistance, R, and capacitance, C, are obtained by combin-ing all the resistors and capacitances at the input of the measuring circuit. In theequivalent circuit it is shown that the total resistance (R) and capacitance (C)

1.2 Signal Induction Mechanism 7

form an RC circuit with a time constant τ = RC. The current pulse induced bythe moving charge carriers on the detector’s electrodes appears as a voltagepulse at the input of the readout electronics. The shape of this voltage pulseis a function of the time constant of the detector circuits. If the time constantis small compared with the duration of charge collection time in the detector,then the current flowing to the resistor is essentially equal to the instantaneousvalue of the current flowing in the detector, and thus themeasured voltage pulsehas a shape nearly identical to the time dependence of the current producedwithin the detector. This pulse is called current pulse. If the time constant islarger than the charge collection time, which is a more general case, then thecurrent is integrated on the total capacitor, and therefore it represents thecharge induced on the electrode. This pulse is called charge pulse. The inte-grated charge will finally discharge on the resistor, leading to a voltage thatcan be described as

V =Q∘

Ce− tτ 1 6

whereQ∘ is the total charge produced in the detector. Because the capacitance Cis normally fixed, the amplitude of the signal pulse is directly proportional to thetotal charge generated in the detector:

Vmax =Q∘

C1 7

Bearing in mind that the total charge produced in the detector is proportionalto the energy deposited in the detector, Eq. 1.7 means that the amplitude of thecharge pulse is proportional to the energy deposited in the detector.

Detector

Bias voltage

Equivalent circuit

PreamplifierRL

Cd Ra

RCVi

Ca=Rd =

Figure 1.5 The arrangement of a detector–preamplifier and its equivalent circuit.

1 Signal Generation in Radiation Detectors8

1.3 Pulses from Ionization Detectors

1.3.1 Gaseous Detectors

The physics of gaseous detectors have been described in various excellent booksand reviews (see, e.g., Refs. [7, 8]). Here only a quick overview of the principles isgiven and more detailed information can be found in the references. Theoperation of a gaseous detector is based on the ionization of gas moleculesby radiation, producing free electrons and positive ions in the gas, commonlyknown as ion pairs. The average number of ion pairs due to a radiation energydeposition equal to ΔE in the detector is given by

n∘ =ΔEw

1 8

wherew is the average energy required to generate an ion pair. Thew-value is, inprinciple, a function of the species of gas involved, the type of radiation, and itsenergy. The typical value of w is in the range of 23–40 eV per ion pair. Theproduction of ion pairs is subject to statistical variations, which are quantifiedby the Fano factor. The variance of the fluctuations in the number of ion pairs isexpressed in terms of the Fano factor F as

σ2 = Fn∘ 1 9

The Fano factor ranges from 0.05 to 0.2 in the common gases used in gaseousdetectors. Under the influence of an external electric field, the electrons andpositive ions move toward the electrodes, inducing a current on the electrodes.If the external electric field is strong enough, the drifting electrons may produceextra ionization in the detector, thereby increasing the amount of induced sig-nal. Depending on the relation between the amount of initial charge released inthe detector and total charge generated in the detector, the operation of gaseousdetectors can be classified into three main regions including ionization chamberregion, proportional region, and Geiger–Müller (GM) region. This classifica-tion is illustratively shown in Figure 1.6. At very low voltages, the ion pairsdo not receive enough electrostatic acceleration to reach the electrodes andtherefore may combine together to form the original molecule, instead of beingcollected by the electrodes. Therefore, the total collected charge on the electro-des is less than the initial ionization. This region is called region of recombina-tion, and no detector is practically employed in this region. In the second region,the electric field intensity is only strong enough to collect all the primary ionpairs by minimizing the recombination of electron ion pairs. The detectorsoperating in this region are called ionization chambers. When the electric fieldis further increased, the electrons gain enough energy to cause secondary ion-ization. This process is called gas amplification or chargemultiplication process.As a result of this process, the collected charge will be larger than the amount of

1.3 Pulses from Ionization Detectors 9

initial ionization, but it is linearly proportional to it. The detectors operating inthis region are called proportional counters. The operation of a detector in theproportional region is characterized by a quantity called first Townsend coef-ficient (α), which denotes the mean number of ion pairs formed by an electronper unit of its path length. The first Townsend coefficient is a function of gaspressure and electric field intensity, and therefore, the operation of a propor-tional counter is governed by the gas pressure and the applied voltage. By havingthe first Townsend coefficient, the increase in the number of electrons driftingfrom location x1 to location x2 is characterized with a charge multiplicationfactor A given by

A= expx2

x1

αdx , 1 10

and the total amount of chargeQ generated by n∘ original ion pairs is obtained as

Q= n∘eA 1 11

From this relation, it follows that the amount of charge generated in thedetector can be controlled by the gas amplification factor, but one shouldknow that the maximum gas amplification is practically limited by the

Applied voltage

Num

ber

of

coll

ecte

d i

on-p

airs

(log s

cale

)

Recombination region

Ionization region Proportional region

Limited proportionalityGeiger–Muller region

Breakdown

Figure 1.6 The classification of gaseous detectors based on the amount of charge generatedin the detector for a given amount of ionization.

1 Signal Generation in Radiation Detectors10

maximum amount of charge that can be generated in a gaseous detectorbefore the electrical breakdown happens. This is called the Raether limitand happens when the amount of total charge reaches to ~108 electrons[9]. Even before reaching to the Raether limit, the increase in the appliedvoltage leads to nonlinear effects, and a region called limited proportionalitystarts. The nonlinear region stems from the fact that opposite to the freeelectrons, which are quickly collected due to their high drift velocity, thepositive ions are slowly moving and their accumulation inside the detectorduring the charge multiplication process distorts the external electric fieldand consequently the gas amplification process. When the multiplicationof single electrons is further increased (106–108), the detector may enterto the GM region. In this regime, the gas amplification is so high that thephotons whose wavelength may be in visible or ultraviolet region are pro-duced. By means of photoionization, the photons may produce newelectrons that initiate new avalanches. Consequently, avalanches extend inthe detector volume and very large pulses are produced. This process iscalled a Geiger discharge. Eventually, the avalanche formation stops becausethe space-charge electric field of the large amount of positive ions left behindreduces the external electric field, preventing more avalanche formation. Asa result, a detector operating in the Geiger region gives a pulse whose sizedoes not depend on the amount of primary ionization. The shape of a pulsefor a gaseous detector depends not only on its operating region but also on itselectrode geometry. In the following sections, we will review the pulse-shapecharacteristics of gaseous detectors of common geometries, operating indifferent regions.

1.3.1.1 Parallel-Plate Ionization ChamberIonization chambers are among the oldest and most widely used types of radi-ation detectors. Ionization chambers offer several attractive features thatinclude variety in the mode of signal readout (pulse and current mode) andextremely low level of performance degradation due to the radiation damage,and also these detectors can be simply constructed in different shapes and sizessuitable for the application. Here, we discuss the pulse formation in an ioniza-tion chamber with parallel-plate geometry, and description of pulses from othergeometries such as cylindrical can be found in Ref. [10].As it is shown in Figure 1.7, the detector consists of two parallel electrodes,

separated by some distance d. The space between the electrodes is filled with asuitable gas. We will assume that d is small compared with both the length andwidth of the electrodes so that the electric field inside the detector is uniformand normal to the electrodes, with magnitude

E =Vd, 1 12

1.3 Pulses from Ionization Detectors 11

where V is the applied voltage between the electrodes. For the purpose of pulsecalculation, we initially assume that all ion pairs are formed at an equal distancex∘ from the anode. In this way, an ionization electron will travel a distance x∘ tothe anode, and a positive ion travels a distance d – x∘ to the cathode. The drifttime Te for an electron to travel to the anode depends linearly on x∘ as

Te =x∘ve

1 13

where ve is the electron’s drift velocity. The ions reach the cathode in a time Tion:

Tion =d−x∘vion

1 14

where vion is the drift velocity of positive ions. The current induced on theelectrodes of an ionization chamber is due to the drift of both electrons andpositive ions. To calculate the current ie induced on the anode electrode dueto n∘ drifting electrons by the Shockley–Ramo theorem, one needs to determinethe anode’s weighting field. The weighting field E∘ is obtained by holding theanode electrode at unit potential and the cathode electrode is grounded. Bysetting V = 1 in Eq. 1.12, E∘ is simply given as

E∘ =1d

1 15

–

+

–

+

–

+

–xº

Anode

Ionizing particle

Cathode

dVout

RL

Gas enclosure

i

High voltage

+

Figure 1.7 The cross section of a parallel-electrode ionization chamber used in deriving theshape of pulses induced by ion pairs released at the distance x∘ from the anode of thedetector.

1 Signal Generation in Radiation Detectors12

Since the directions of the electrons’ drift velocity and the external electricfield are opposite, Eq. 1.4 gives the current induced by the electrons on theanode as

ie t = − −n∘e −ve1d= −

n∘eved

0 < t ≤Te 1 16

The negative sign of n∘e is due to the negative charge of electrons. Once anelectron reaches the anode, it no longer induces a current on the anode andtherefore ie = 0 for t > te. Equation 1.16 indicates that the polarity of the pulseinduced on the anode is negative, which is in accordance with the rule thatwe mentioned in Section 1.2.1. If we calculate the current induced on the cath-ode by electrons, the drift velocity of electrons and the weighting field are in thesame direction, and thus, the polarity of induced charge will be positive. Theinduced current by positive ions on the anode can be similarly calculated as

iion t = −n∘e vion

d= −

n∘eviond

0 < t ≤Tion 1 17

The total induced current on the anode is a sum of contributions fromelectrons and positive ions, given by

i t = ie t + iion t = −n∘eved

−n∘evion

d= −

n∘ed

ve + vion 1 18

The top panel of Figure 1.8 shows an example of induced currents on theanode of an ionization chamber. The figure shows a hypothetical case in whichthe drift velocity of electrons is only five times larger than that of positive ions.In practice, the drift velocity of electrons is much larger than positive ions(~1000 times), and thus the induced current by positive ions has much smalleramplitude and much longer duration. The calculated induced currents haveconstant amplitude because of the constant drift velocity of charge carriersand have zero risetimes though this cannot be practically observed due tothe finite bandwidth of the detector circuit. The charge pulse induced on theelectrodes as a function of time can be obtained by using the Shockley–Ramotheorem (Eq. 1.5) or alternatively by a simple integration of the calculatedinduced currents. The integral of ie(t) over time, which we denote it as Qe(t),represents the induced charge on the anode due to the n∘ drifting electrons as

Qe t =Te

0iedt = −

n∘ed

vet 0 < t ≤Te 1 19

The polarity of this pulse is opposite to the polarity of induced charge, whichis obtained from Eq. 1.5. This is due to the fact that the Shockley–Ramo theoremgives the total induced charge on the electrode, while the integration of currentpulse represents the outgoing charge from the electrode or the observed pulse.The induced charge increases linearly with time until electrons reach the anode

1.3 Pulses from Ionization Detectors 13

after which the charge induced by electrons remains constant. Similarly, theinduced charge Qion(t) by the drift of positive ions is given by

Qion t =Tion

0iiondt = −

n∘ed

viont 0 < t ≤Tion 1 20

The positive ion pulse also linearly increases with time, but with a smallerslope due to the smaller drift velocity of positive ions. The total induced chargeon the anode, during the drift of electrons and positive ions, is obtained as

Q t =Qe t +Qion t = −n∘ed

ve + vion t 1 21

After the electrons’ collection time, Te, the electrons have contributed to themaximum possible value, and the electron contribution becomes constant. Butif the positive ions are still drifting, Eq. 1.21 takes the form

Q t = −n∘ed

x∘ + viont 1 22

When both the electrons and ions reached their corresponding electrodes,Eq. 1.22 is written as

Q t = −n∘ed

x∘ + d−x∘ = −n∘e 1 23

Ion current

Electron current

Electron collection

time

Ion collection time

Time

Time

Ind

uce

d

curr

ent

Ind

uce

d

char

ge

nºe

Figure 1.8 (Top) Time development of an induced current pulse on the anode of a planarionization chamber by the motion of electrons and positive ions. The figure is drawn as if theelectron drift velocity is only five times faster than the ion drift velocity. (Bottom) The inducedcharge on the anode.

1 Signal Generation in Radiation Detectors14