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Performance of a position sensitive low-pressure wire chamber (LPWC) having position readout from a separate sense wire plane: a critical analysis
Nuclear Instruments and Methods in Physics Research A 484 (2002) 407–418
Performance of a position sensitive low-pressure wire chamber (LPWC) having position readout from a separate sense wire plane: a critical analysis Chinmay Basu*, B.P. Das, Subinit Roy, P. Basu, H. Majumdar, M.L. Chatterjee Saha Institute of Nuclear Physics, 1/AF, Bidhan Nagar, Kolkata 700064, India Received 19 June 2000; received in revised form 26 July 2001; accepted 17 September 2001
Abstract The paper describes the operation of a simple one-dimensional LPWC from a completely different standpoint. The position output is derived from a separate position plane, where discrete resistances are soldered between two adjacent wires. Theoretical formulations for the position extraction are derived where the role of various detector parameters are dealt with. Position spectrum is shown for various choices of resistances varying from 50 O to 220 kO: Best position resolution and linearity is obtained for 33 kO though it violates the existing hypothesis. r 2001 Elsevier Science B.V. All rights reserved. PACS: 29.40.Cs; 29.40.Gx Keywords: Wire chamber; Position-sensitive; Resistive charge-division
1. Introduction Low-pressure position sensitive multiwire proportional counters (LPMWPC) have been widely used to detect heavy ions for their excellent position resolution and ultrafast timing properties [1–4]. A brief review of these types of detectors is given in Ref. [5]. The LPMWPC is a proportional counter with three electrodes viz., an anode wire plane placed between two cathode planes parallel to one another. MWPCs operating at relatively *Corresponding author. Tel.: 091-033-3374321; fax: 091-0333374637. E-mail addresses: [email protected] (C. Basu), [email protected] (P. Basu).
higher gas pressures (> 10 Torr) were never considered as time measuring devices because their time resolutions are typically > 50 ns: This is due to the long drift time of the electrons, released in the sensitive volume by the penetrating ionizing radiation, in reaching the anode wire, where amplification occurs. However the amplification mechanism for low-pressure counters is entirely different. The reduced electric field strength E=p (E ¼ potential difference per unit length in the sensitive volume and p ¼ pressure of the gas within the detector volume), in the constant field region, reaches a value of the order of several hundred V cm1 Torr1 : Primary electrons are accelerated in this strong uniform electric field between the electrodes, which causes ionization of
0168-9002/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 2 0 3 4 - 4
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additional atoms. Successive ionization will result in the formation of the so-called Townsend avalanche. A primary electron created near the cathode is multiplied by several thousand before it reaches the vicinity of the anode wire. Close to the wire, the electric field strength increases even more strongly (D1=r) leading to a second electron multiplication in its vicinity. The total gain of such LPMWPC, due to this two-stage amplification process is typically 106 –107 : Thus output pulses have large amplitude, especially for heavy charged particles. Due to this strong electric field the drift time of these electrons are reduced considerably and one easily obtains rise times of the order of 1–5 ns: Thus, a time resolution of several hundred picoseconds (FWHM) can be reached under these conditions, depending on the size and geometry of the device. As the spatial distribution of each avalanche is localized around each anode wire (within 71–2 times the diameter of the wire), a method to derive the position and/ or angular information of the incident charged particle can be easily realized. Several methods of extracting the position information have been described in literature viz., center of gravity method, delay line technique, charge division method, etc. In the ‘delay line’ method, fixed delays (B2–5 ns) are introduced between each successive cathode or sense-wire plane. If the induced pulses are viewed from one end, then depending on the point of impact it will see different delays. This is an obvious approach because of the accuracy and simplicity of this type of readout methods [6]. The second method of position determination utilizes the resistive electrodes. This might be the resistive anode plane or, a separate position readout plane between the anode and cathode. The distributed resistance of this plane and the distributed capacitance between the anode and cathode make the LPMWPC equivalent to a distributed RC line. Radiation striking the detector injects a charge impulse at the point of incidence and both the amplitude and rise time of the current pulses observed at each end of the counter are dependent on the position of incidence. Thus in this method, again there are two ways of position determination, viz., the ‘‘rise time method’’ and the ‘‘charge division method’’. An
elaborate discussion about these methods can be found in Refs. [7,8], respectively. In the charge division method, position is determined by the ratio of the charge flowing out at one end of the resistive electrode, terminated into low impedance, and the total charge. The position information can be obtained either, by the evaluation of the centroid of this charge distribution [9], or a relatively simpler method is to use a global readout technique [10]. The latter method measures the charge output at the two ends of this resistive electrode. In the present work, we use a simple variant of the classical LPMWPC; with only one cathode plane instead of two. Both the cathode and anode are in the form of thin wire planes. The electric field distribution for this structure is similar to a MWPC and the mode of operation involves twostage amplification as described for a LPMWPC. However, owing to its difference in design from a classical MWPC we shall refer the present detector in this article as a low-pressure wire chamber (LPWC). In order to obtain position information, an additional wire plane is inserted between the anode and cathode. Eyal and Stelzer used a similar structure in the past for a two-dimensional PPAC [11]. The method of resistive charge division has been adopted in this work for determination of position. We have tried to characterize the LPWC with discrete resistance readout, using the global readout technique. This technique reduces complications in the readout electronics and proves to be more cost effective. This is more important in the present context with the rapid development and usage of large area detectors and multi-detector arrays. Moreover, we use the charge dividing resistors on the position plane, instead of the more widely used resistive anode charge division [12] or cathode strip readout [9]. Though earlier workers [8,10,13,14] have performed extensive studies on RC encoding, the discussions were confined more towards optimization with regard to linearity and resolution, in terms of theoretical noise analysis and optimum filtering. The effects of practical deviations occurring in a real (operating) position sensitive detector, from the commonly assumed point injection of charge has been less considered.
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In this work, we study resistive charge division by two-channel readout with equal emphasis on theory and experiment. An elaborate discussion into the theoretical aspects of the LPWC has been included wherein the role of various parameters have been critically examined. Optimization of various parameters like resistors, pressure of the gas, reduced electric field and shaping time of the amplifier have been carried out in conjunction with the theoretical interpretations. A position resolution of less than a millimeter has been obtained inspite of the crudity of the global readout method. The paper is organized as follows. In Section 2 we describe the construction of the detector. The consistency of the detector behavior has been analyzed with the existing theoretical in terpretations in Section 3. Experimental details and discussion of the results are given in Section 4. Finally, summary and conclusions are given in Section 5.
2. Construction of the detector The constructional details of the counter are shown in Fig. 1. The cathode and anode wire planes are made of 12:5 mm thick gold-plated tungsten wires soldered on a copper clad G-10 board at a spacing of 1 mm: Use of thin wires in the cathode plane is a source of field emission. This problem is averted by the use of thin conducting foil [3]. Our primary objective however, of fabricating the present detector, is to use it in the focal plane of a magnetic spectrograph for efficient measurement of low energy, high Z evaporation residues. Use of metallized foils would reduce the
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transmission drastically. To extract position information, a separate plane is inserted (Fig. 1) between the anode and cathode, with wires thin enough, to minimize distortion of field lines. The sense wires are 25 mm in diameter and spaced 2:5 mm apart. Discrete resistors are soldered between each adjacent sense wire. The value of the resistors used in our experiment varied between 50 O and 220 kO: Two such prototype detectors, having areas 90 65 and 75 50 mm2 were used. The distance of separation between cathode and anode wire planes were varied between 7 and 10 mm; whilst that between sense wire and anode is kept fixed at 3 mm: The gas windows are made of 50 mg cm2 stretched polypropylene film. A primary objective of using large area position sensitive gas counters is to utilize its very high rate handling capacity (B106 counts s1 ). However, charge division method is inherently slow compared to other methods of position determination, viz., the delay line or the center of gravity method. A possible way of eliminating this problem is to place the position plane close enough to the anode for fast and efficient position determination. Normally, one uses the cathode plane to extract position information. It is our experience, however, that there is always a problem of sparking when the anode–cathode gap is made small and stable operation of the counter for extended periods become impossible. Besides, the separate sense plane becomes useful in obtaining twodimensional readout if the cathode is also used as a position plane [11]. The detector was operated with 99% pure isobutane gas at a pressure varying between 2– 5 Torr: The pressure inside the detector volume
Fig. 1. The schematic diagram of the low pressure position sensitive wire counter (LPWC). The anode A (12:5 mm gold-plated tungsten wire, 1 mm apart), the cathode C (12:5 mm gold-plated tungsten wire, 1 mm apart) and the sense wire plane S (25 mm gold-plated tungsten wire, 2:5 mm apart) are shown. The wires are soldered on G-10 boards. The gas window and supporting mesh are not shown. Typical voltages applied on the anode and cathode wire planes are also shown.
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was kept fixed during the experiment by means of an electronic pressure-regulating valve. A well-collimated 252 Cf fission source (strength 10 mCi) was used during testing of the LPWC. The source to detector distance was kept at 10 cms: The typical anode pulse characteristics for our detector and also for an equivalent LPMWPC are given in Table 1. In spite of the simple design of the present chamber, the pulse height and rise times are of the same order as in a LPMWPC. The schematic layout of the electronics is shown in Fig. 2. The charge signals from the two ends of Table 1 Typical anode pulse characteristics of fission fragments detected from a 252 Cf source with the detector described in this work and a LPMWPCa Anode pulse characteristics
Anode–cathode gap (mm)
LPWC (present detector)
LPMWPC
Pulse height
4.0 8.0
500 mV 700 mV
1V 1:5 V
Rise time
4.0 8.0
4 ns 7 ns
4 ns 7 ns
a The operating pressure was kept at 3 Torr with E=P around 250 V cm1 Torr1 : The pulse amplitude and risetimes obtained after a VT120, Gain 200, ORTEC fast preamplifier and recorded in a 500 MHz HP Oscilloscope are depicted in the table. The noise level and the approximate gain of the present detector are also shown. Noise ¼ 15–20 mV; Gain (LPWC)B1056 :
Fig. 2. Simplified detector and electronic circuit for position measurements with the LPWC using the charge division technique.
the sense wire planes were taken through conventional charge sensitive preamplifiers (ORTEC 142IH). They were subsequently shaped and amplified through active filter amplifiers (ORTEC 572) and summed to give the total charge. The ratio of one end of the signal to the total gives the position information. Before the experiment, the preamplifiers and amplifiers were properly gain matched by a research pulser. The anode and cathode volt ages were given through ORTEC bias supplies.
3. Mechanism of charge division in the low-pressure wire chamber The principles regarding the operation and characteristics of the basic proportional counter and the development of pulses within these detectors have been extensively reviewed [15]. However, essential features of the theory of operation of the present counter as a position sensitive device will now be discussed in order to provide a physical understanding of their properties. The mode of operation of the present lowpressure wire chamber (LPWC) is similar to that of a conventional low-pressure MWPC. In the LPWC, a two-stage amplification of the electrons take place, which essentially depends on the reduced electric field E=p and the gas used. The electric field is constant in the drift region (at distances greater than several times the anode wire radii). However, the value of E=p here is of the order of several hundred V cm1 Torr1 ; and is sufficient for secondary ionization. Nearer to the anode, the field varies as 1=r- (see appendix) and a second stage of amplification sets in. This latter amplification of electrons occurs very close to the anode wire and is localized in space. They give rise to an ultrafast anode signal (Bns). The purpose of the position readout is to sense these avalanches. The avalanches created near the anode will induce a charge signal on the nearby sense wire. The induced charge will flow out at the two ends of the position readout plane with a magnitude dependent on the total resistance faced in its path. If we consider this resistive electrode as an effective RC transmission line, with distributed R and C per
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unit length, the charge division and diffusion in one dimension can be treated through Telegrapher’s equation [12] (with G (conductance) and L (inductance) being negligible):
Considering a point injection of charge Q at x ¼ x0 (avalanche point) and at time t ¼ 0; we have in terms of the Dirac d-function, dðx x0 Þ;
@2 uðx; tÞ @uðx; tÞ ¼ RC @x2 @t
uðx0 ; 0Þ ¼
ð1Þ
where uðx; tÞ is the voltage induced on the line at position x and time t; R the resistance per unit length and C the capacitance per unit length. Eq. (1) is a standard diffusion equation in onedimension. It can be solved by means of separation of variables X uðx; tÞ ¼ Xn ðxÞTn ðtÞ: ð2Þ
Q dðx x0 Þ: C
ð8Þ
Substituting Eq. (8) in Eq. (7) and using Fourierseries expansion, we finally arrive at N npx npx 2Q X 2 2 2 0 uðx; tÞ ¼ sin eðn p t=RCl Þ : sin Cl n¼1 l l ð9Þ
Substitution of Eq. (2) into Eq. (1) gives
In order to arrive at the expression for current from Eq. (9), we use the first coupled Telegrapher’s equation
1 d2 Xn ðxÞ RC dTn ðtÞ ¼ l2n ðconstantÞ: ¼ Xn ðxÞ dx2 Tn ðtÞ dt
@iðx; tÞ @uðx; tÞ þC ¼0 @x @t
n
ð3Þ Solving for the time and position part separately gives 2
Tn ðtÞ ¼ Tn ð0Þeðln t=RCÞ
wherefrom iðx; tÞ ¼
npx npx 2pQ X 0 n sin cos RCl 2 n l l
ð4Þ 2 2
eðn p
and, Xn ðxÞ ¼ An sinðln xÞ:
ð5Þ
For the latter, we have assumed both ends of the line being terminated by zero impedance preamplifier, i.e. uðx ¼ 0; tÞ ¼ uðx ¼ l; tÞ ¼ 0; where l is the length of the line. From Eq. (5) this gives np ln ¼ : ð6Þ l Therefore, we finally arrive at N npx X 2 2 2 uðx; tÞ ¼ Cn sin eðn p t=RCl Þ ; l n¼1
ð10Þ
ð7Þ
where An and Tn ð0Þ are absorbed in Cn : The above expression signifies the voltage distribution along a one-dimensional distributed RC-line, for any point x along the length of the line and at any finite time t: The solution consists of a sinusoidal space part and an exponentially decaying time part, both of which depends on the line parameters, viz., R; C and l:
t=RCl 2 Þ
:
ð11Þ
From the above expression the values of the current, i; at the two ends, i.e., x ¼ 0 and l will be npx 2pQ X 2 2 2 0 n sin ið0; tÞ ¼ eðn p t=RCl Þ 2 RCl n l and iðl; tÞ ¼
npx 2pQ X 2 2 2 0 n sin cosðnpÞeðn p t=RCl Þ : 2 RCl n l
Since Qð0Þ þ QðlÞ ¼ Q; is a conserved quantity, we define a convention that Qð0Þ decreases with increasing x and QðlÞ increases with increasing x: Therefore, we arrive at Z t ið0; t0 Þ dt0 Qð0; tÞ ¼ 0
¼
2Q X sinðnpx0 =lÞ 2 2 2 ð1 eðn p t=RCl Þ Þ p n n ð12Þ
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and Qðl; tÞ ¼
Z
t
iðl; t0 Þ dt0 0
¼
2Q X sinðnpx0 =lÞ cosðnpÞ p n n 2 2
ð1 eðn p
t=RCl 2 Þ
Þ:
ð13Þ
The time varying part of the charge expressions (12) and (13) are averaged over a time equal to the amplifier shaping time (tF ) by the appropriate weighting function of the amplifier. Alberi and Radeka [8,10] carried out similar averaging, though for a trapezoidal weighting function. In our case we have used ORTEC 572 amplifiers, which uses an approximation to Gaussian shaping. Incorporating this weighting function into the above equation and simplifying we finally obtain x0 px0 Q1 ¼ Qð0; tF ÞDQ 1 ð14Þ QK sin l l Q2 ¼ Qðl; tF ÞDQ
x0 px0 QK sin l l
ð15Þ
px0 Q1 þ Q2 ¼ Q 2QK sin ð16Þ l where K is the non-linearity term which for a semiGaussian weighting function is obtained as eð1XÞ 2 2e 1 X þ K¼ ð17Þ 1 X X2 X X where X ¼ ðp2 tF =tD þ 1Þ; tD being time constant of the line, i.e. tD ¼ RCl 2 : Thus, using a finite integration time, a nonlinearity K is introduced which is a function of tF =tD : Ideally, for integration time tF -N; K-0; and the asymptotic charge expression at the two ends of the line becomes 2Q X sinðnpx0 =lÞ x0 ¼Q 1 Q1 ¼ Qð0Þ ¼ ; p n n l ð18Þ where the final equality results from the standard summation of the series. Similarly from Eq. (15), using tF -N; we arrive at the expression for the other end Qx0 Q2 ¼ QðlÞ ¼ : ð19Þ l
The position signal is obtained by taking the ratio of charge at one end to the total charge, i.e., Q1 =Q1 þ Q2 or Q2 =Q1 þ Q2 : It is easily seen from Eqs. (18) and (19) that position signals are strictly proportional to x0 =l (distance of avalanche from one end). However, one must keep in mind that we have taken the asymptotic charge expression (tF -N). This is an ideal situation not realized in practice. Let us now consider how the situation modifies for finite shaping times (Bms for spectroscopy amplifiers). The position signal for finite tF can be constructed from Eqs. (15) and (16):
Q2 x0 2x0 px0 : ð20Þ E K 1 sin Q1 þ Q2 l l l Thus, there is always some non-linearity associated with finite tF ; depending on K: We now define the dynamic range of the spectrum, D as
Q1 Q1 D¼ ð21Þ Q1 þ Q2 max Q1 þ Q2 min in terms of the maximum and minimum extent of the position spectrum. Ideally, for tF -N; D ¼ 1: But for finite shaping time, we can write the dynamic range (after some algebraic manipulation of Eq. (20)) as k1 pk p 1 þ K sin þ sin DE ð22Þ kþ1 kþ1 kþ1 where k is the number of sense wires used. It can be easily seen from Eqs. (22) and (17) that the dynamic range increases with K; i.e. for larger tD (high resistance). In fact this effect may result in D exceeding unity for a large value of resistance. It can also be visualized that with fixed tD ; if we increase the shaping time, tF ; of the amplifier, the dynamic range decreases. This can lead to squeezing of the spectrum at higher shaping times. Thus, a proper choice of the value of resistance used and the shaping time of the amplifier circuit is of prime importance. In the above discussion, we have assumed a point injection of charge (Eq. (8)) at x ¼ x0 and t ¼ 0: However, for our detector configuration, we have a separate sense wire plane at a distance ‘L’ from the anode plane (position of actual avalanche) and the individual sense wires are
C. Basu et al. / Nuclear Instruments and Methods in Physics Research A 484 (2002) 407–418
separated by a distance ‘s’. This configuration will lead to a charge distribution in the sense plane, and the previous point charge hypothesis has to be modified. A charge distribution, used earlier [16] would be appropriate Q pðxi x0 Þ qðxi Þ ¼ sec h ð23Þ 4L 2L where x0 is the position of actual avalanche at anode, xi is the position of ith sense wire form one end and L the distance of separation of anode and sense wire plane. Considering a linear charge distribution one can write the ‘position’ or the charge ratio, in terms of the charge distribution as Rl qðxÞðl xÞ dx Q1 ¼ 0 Rl Q1 þ Q2 qðxÞl dx 0
Rl ¼ Z
0
qðxÞðl xÞ dx ; Q:l
ð24Þ
l
qðxÞ dx ¼ Q; 0
where qðxÞ is the linear density of charge. The above expression is valid for a continuously varying resistive plane. For a readout from a sense wire plane, as in our case, with discrete number of wires, we can modify Eq. (24) to read Pn Q1 qðxi Þðl xi Þ ; ð25Þ ¼ i¼1Pn l i¼1 qðxi Þ Q1 þ Q2 where ns ¼ l; n being the total number of sense wires. Using the charge distribution (23) for the sense wire, and applying it to Eq. (25), the behaviour of the position signal (charge ratio) on L and s can be examined. Eq. (25) essentially signifies that though the avalanche is localized on one particular anode, the charge ratio Q1 =Q1 þ Q2 is actually obtained by summing contributions from all sense wires. The magnitude of charge injected on each sense wire (a fraction of Q) gets progressively reduced as we go away from the point of localization. That is, for a sense wire plane kept at some distance away from the anode plane (plane of localization), the mechanism of charge division is different from that of point injection of charge.
413
In the preceding discussions, we have briefly reviewed the theory of charge diffusion for point charge injection. Non-linearity in the position extraction has been derived for a semi-Gaussian weighting function and is seen to depend on tF =tD : Some simple calculations were also done to treat the mechanism of charge division on a separate position plane. For our detector configuration, it is a charge distribution that one has to deal with instead of point charge injection. How these factors affect the detector resolution and linearity will be considered in the next section.
4. Results and discussions The primary motive of the experiment is to find out proper choice of resistance, (r), used between adjacent sense wires such that the position resolution and/or linearity is optimum. In order to arrive at a definite conclusion we have used various discrete r between each successive sense wire. The values of r used ranged from 50 O to 220 kO: Our detector dimension could accommodate a maximum of 30 resistors for the smaller detector (75 mm 50 mm) and 36 for the larger one (90 mm 65 mm). The capacitance of these prototype detectors were also measured and found to be B20 pF: Thus the RC-value (denoted by tD ; the time constant of the resistive circuit) of sense wire line for each r can easily be calculated and are tabulated in Table 2. We have measured the position resolution for each choice of resistance value. This was done by uniformly illuminating a mask, having 1 mm wide slits placed 1 mm apart. The pressure of the isobutane gas within the detector volume was kept fixed between 3 and 5 Torr: The reduced electric field E=p was varied between 150 and 350 V cm1 Torr1 : Bias voltages were applied on both the anode and cathode planes while keeping the sense wire plane floating. The schematic diagram of the measuring circuit is shown in Fig. 2. The charge outputs from the two ends of the RC-line were fed to a charge sensitive preamplifier (ORTEC 142IH in our case) and then shaped and amplified in a spectroscopy amplifier (ORTEC 572 in our case) which provides
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Gaussian filtering (CR-RC6 ). They are further processed in a dual sum and invert (ORTEC 533) and analog divider (ORTEC 464), properly calibrated to ensure non-linearity within 0:1%: All earlier studies on RC-encoding by charge division was made from a more theoretical viewpoint using idealization such as point charge injection. Gas counters operating at higher pressures have only been considered and ratio of filtering time constant (tF ) to the RC line time constant (tD ) was shown to be an important factor [8,10,13]. So in this work, the experiments were performed at different amplifier shaping times for each choice of r: We have shown in Table 3, values of the parameter tF =tD ; for various choices of r and available settings of amplifier shaping times ðtF Þ: The values of tF =tD are very high for lower r and decreases as we progressively increase the value of resistance. Fig. 3 shows the results of the experimental position spectrum for various r and tF : The values Table 2 Value of RC (denoted by tD ) for different combinations of resistors and detector dimensions. The capacitance of the detector measured from end to end is B20 pF r (in O)
tD (in ms)
Detector 1 ð90 mm 65 mmÞ 36 resistors
50 100 200
0.036 0.072 0.144
Detector 2 ð75 mm 50 mmÞ 30 resistors
680 4:7 103 10 103 33 103 220 103
0.4 2.82 6.00 19.80 132.00
of the factor tF =tD are also shown within parentheses for each figure. The different choices were selected to represent the best spectrum for each resistance value. If we scan through the different choices (Fig. 3(a)–(f)), it can easily be surmised that two values r ¼ 680 O (Fig. 3(b)) and r ¼ 33 kO (Fig. 3(e)) are the better choices. Clearly separated peaks and excellent position resolution (o1 mm FWHM) are achieved. The peaks also rise from the baseline showing negligible component of background noise. For the 33 kO case the intensity distribution is also in consonance with placement of a well-collimated source at the mid-point of the detector area. For other choices of r; some proportion of noise is always present. In Table 4, we have given for some representative cases the position resolution (FWHM) with reference to 1 mm slit and the limit to resolution in each case due to electronic noise. To determine the latter, equal charge from a pulser were fed to the two preamplifier test inputs at the two ends of the RCline which was kept floating. The position signal FWHM obtained is indicated in units of mm. One interesting aspect of the experiment is that in general we get fairly good resolution (o1 mm FWHM) over a wide range of tF =tD (see Table 4), whereas earlier workers [8,10,13] concluded that good resolution is expected only when tF =tD 51: For r ¼ 100 O (Fig. 3(a)) and other low values of r below 680 O there is significant amount of noise for all values of amplifier shaping times. However, it appears that the position resolution is better at lower values of tF =tD (Table 4), similar to the trend predicted by Refs. [8,10,13]. Fig. 4 is a representation of the position linearity. Shown here are the distance of the
Table 3 Value of the parameter tF =tD for different choices of resistors used and also for available settings of amplifier time constants (tF ) Shaping time tF ðmsÞ 0.5 1.0 2.0 3.0 6.0 10.0
tF =tD 50 O
100 O
200 O
680 O
4:7 kO
10 kO
33 kO
220 kO
13.9 27.8 55.6 83.4 166.8 278.0
7.0 13.9 27.8 41.7 83.4 139.0
3.5 6.95 13.9 20.85 41.7 69.5
1.22 2.44 4.88 7.32 14.64 24.40
0.175 0.348 0.695 1.043 2.085 3.475
0.083 0.167 0.333 0.501 1.002 1.67
0.025 0.050 0.101 0.151 0.303 0.505
0.0038 0.0080 0.0150 0.0230 0.0450 0.0760
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C. Basu et al. / Nuclear Instruments and Methods in Physics Research A 484 (2002) 407–418
(a)
600
τF=0.5 µs r =100 Ω
τF=0.5 µs, r =680 Ω
80
(7.0)
400
(b)
100
(1.22)
60 40
200
20 0
0 250
750 1000
0
250 500 750 1000
(c)
400
Counts
500
300
(d)
50
τF=2 µs r =4.7 kΩ (.695)
τF=3 µs r =10kΩ (.501)
40 30
200
20 100
10
0
centers of various slits (uniformly spaced) against each peak channel number (y-axes). Ideally for precise position information it should strictly fall on a straight-line fit (linear regression [17]). The goodness of fit may be quantified by the correlation coefficient c; of the regression line. This is indicated in Figs. 4(a)–(f) within brackets. Closer the value of c to unity, better is the fit and hence the linearity. However, we see that there are small oscillations about the mean straight line. These minor variations represent the differential nonlinearity associated with each setting. A striking feature of the non-linearity is that they are oscillatory in nature. This is understood from the
0 250
500
750 1000
(e) τF=1 µs r =33 kΩ
100
(.05)
80
600
250 500 750 1000
(f)
120
1000 800
0
(a) 2500
τF=10 µs r =220 kΩ (.076) (.076)
2000
200 0
40
1000
20
500
250 500 750 1000
220kΩ,τF=6 µs
τF/τD=0.0075
1500
τF/τD=0.045
1000 500 (0.993)
(0.971)
0 0
220kΩ,τF=1 µs
1500
60 400
(b) 2000
0
0 0
250 500 750 1000
0 10 20 30 40 50 60
0 10 20 30 40 50 60
Channle
(c) 1200
Peak Channel
Fig. 3. (a)–(f) Position spectrum obtained for various combinations of tF and r: The numbers within the brackets are the ratio tF =tD : The spectrum is obtained by uniformly illuminating a rectangular mask having 1 mm slits and 1 mm opaque region.
1000 800
(d )
33kΩ,τF=1 µs
2000
τF/τD=0.05
1600
680Ω,τF=0.5 µs τF/τD=1.22
1200
600
800
400 200
(0.997)
0
400
(0.999)
0 0 10 20 30 40 50 60
0 10 20 30 40 50 60
(f)
(e) Table 4 Experimental position resolution (FWHM) expressed in millimeter for different choices of tF =tD . Value of noise (FWHM) is also indicated in millimeter tF =tD 0.050 0.076 0.083 0.150 0.500 1.220 7.000 7.320 24.400
r ðOÞ 33 k 220 k 10 k 33 k 33 k 680 100 680 680
tF ðmsÞ 1.0 10.0 0.5 3.0 10.0 0.5 0.5 3.0 10.0
FWHM (mm) 0.757 0.779 0.797 0.857 0.925 0.748 1.75 0.842 0.902
Noise FWHM (mm) 0.360 0.104 0.070 0.180 0.230 0.430 0.940 0.150 0.180
1200
1800 1600
680Ω,τF=10 µs
1400
τF/τD=24.4
1000
100 Ω, τF=0.5,2 µs
800
1200
600
1000
400
800
200
(0.999)
τF/τD=7.0 τF/τD=27.8 (0.993,0.987)
0
600 0
10 20 30 40 50
0 10 20 30 40 50 60 70
Position (mm) Fig. 4. (a)–(f) Linearity plots (peak channel vs. position) obtained for various choices of tF and r: Values of tF =tD are also indicated. The experimental points are indicated by dots. The straight-line fits (linear regression) are also shown. The correlation coefficients of the regression line are indicated within brackets.
C. Basu et al. / Nuclear Instruments and Methods in Physics Research A 484 (2002) 407–418
position term in Eq. (20) for finite tF ; derived for point charge injection. The oscillations should have intercepts at 0, l=2 and l on the straight line. In our case the source has an extended charge distribution at the position plane and so the oscillations show additional intercepts in some cases (Fig. 4(b)). In Section 3, assuming point charge injection, we derived the non-linearity factor K for the Gaussian filter (Eq. (17)) which was found to be decreasing with increasing values of tF =tD : This trend is also reproduced in our experimental observation in Fig. 4, except for r ¼ 100 O (Fig. 4(f)). However, we get acceptable experimental linearity (c ¼ 0:993; 0:997) even when tF is only 5% of tD (Fig. 4(b) and (c)) where K from Eq. (17) has a large value of about 0.7 (linearity 0.3). This indicates that K needs to be redefined for extended charge distribution. To make a more stringent choice of r; we further compare in Fig. 5 the position spectrum for 680 O and 33 kO; at three different shaping times. It is clearly seen that the dynamic range of the position spectrum gets progressively reduced for higher tF ; predominantly for r ¼ 680 O: The reduction for the other resistance value r ¼ 33 kO is minimal. At the highest shaping time (6 ms), the range for 680 O is already 30% reduced compared to 33 kO: Therefore, we make two significant observations: (a) For a particular choice of r; the dynamic range gets smaller with increasing tF and (b) The squeezing effect is more pronounced for smaller r: Effect (a) can be understood with reference to Eq. (17), where for a particular tD with increasing tF ; K decreases, thereby decreasing D: Effect (b) is a result of faster decrease of K with the increase of tF =tD for 680 O; in comparison to r ¼ 33 kO:
5. Summary and conclusions The paper describes the operation of a simple low-pressure wire chamber from a different stand-
τF/τD 600
τF=1µs
r =33 kΩ
0.05
300 100 r =680 Ω
60
τF=1µs
40
2.44
20 0 0
500
600
Counts
416
1000
1500
2000 τF=3µs
r =33 kΩ
0.151
300 100 60
τF=3µs
r =680 Ω
40
7.32
20 0 0
500
1000
1500
2000 τF=6µs
r =33 kΩ
150
0.303 50 60
=6µs ττFF=6µs
r =680 Ω
40
14.64
20 0 0
500
1000
1500
2000
Channel Fig. 5. Comparison of position spectrum for two choice of resistance values, r ¼ 680 O and 33 kO; shown for different shaping times (tF ¼ 1; 3; 6 ms). The value of the ratio tF =tD is given for each spectrum.
point. In our detector configuration, the position grid is a separate wire plane with discrete values of resistors between two adjacent wires. Position resolution and linearity have been measured over a wide range of resistance values (50 O–220 kO). The role of the finite shaping time of the amplifier in the proper choice of resistance value has been considered. Finally, theoretical insights into the dependence of various parameters, both for the ideal case (tF -N) and real situation (tF finiteEms) are also provided. Theoretical aspects of charge division on a separate wire plane
C. Basu et al. / Nuclear Instruments and Methods in Physics Research A 484 (2002) 407–418
considering spatially extended charge distribution has also been discussed. The conclusions can be classified broadly for (a) Lower values of resistors: The RC-line noise becomes an important criterion. This worsens the position resolution. The dynamic range becomes small when r is less than or close to the preamplifier input impedance. Linearity is affected. A strong squeezing effect is also observed in the position spectrum with increasing amplifier shaping time. (b) Higher value of resistors: The line noise and squeezing effect gets considerably reduced. Measured position resolution is less than a millimeter over a wide range of amplifier shaping times. The observed position linearity is acceptable even at a low value of the ratio of the filtering time constant to the RC-line time constant. A deviation from the criterion for optimum linearity and resolution (tF =tD > 0:5) deduced earlier [8,10] is thus observed. To account for this discrepancy, redefinition of the theoretical non-linearity considering (a) extended charge distribution and (b) pure Gaussian filter is suggested. A detailed theoretical calculation will be extremely complicated and may be pursued in future. We finally conclude that higher resistance values are preferable and consider r ¼ 33 kO to be the optimum choice for our detector design. The present study provides an extension of earlier results [8,10,13,14] on RC encoding. A specific analysis has been carried out that has not been probably done before, viz. the response of a RC electrode to a spatially extended charge injection in a low-pressure wire chamber.
Acknowledgements The authors thankfully acknowledge the kind interest of Professor Sudip Ghosh in the present work and the support and encouragement received from him.
417
Appendix A The electric field distribution for a MWPC of classical design with two cathodes and the anode in between is well known [18,19]. The detector in the present work has only one cathode placed parallel to the anode (Fig. 1). The basic electric field distribution is however similar to that of a standard MWPC. The potential at any point z in the complex plane due to a plane of uniformly charged parallel thin wires in front of a conducting plane cathode (an equivalent image structure is that of the present detector structure) is given by Ref. [19] as FðzÞ ¼ 2Q ln
sin pðz iLÞ=s ; sin pðz þ iLÞ=s
where Q is the charge per unit length and is given as V0 =f2pðL=sÞ 2lnðpd=sÞg; in terms of the anode potential V0 ; anode–cathode gap L; wire diameter d and inter wire separation s: The general expression for electric field is given by
@
jEj ¼
FðzÞ
@z
2pQ
pðz iLÞ pðz þ iLÞ
¼ cot cot
: s s s A.1. Field away from the anode wire With the origin of the z plane on the cathode, this region is where jzj5L: The electric field can then be written as
2pQ
pðiLÞ pðiLÞ
jEjE cot cot s s s 4pQ pL 4pQ coth E : s s s The last expression can be written, as cothðpL=sÞ tends to 1 as pL=sb1: Thus, the electric field is constant away from the anode wire. ¼
A.2. Field close to the anode wire In this region it is convenient to use the ðr; WÞ definition of z as in Ref. [19], i.e.,
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z ¼ iL þ r expðiWÞ; where r is measured from the wire as origin. Using this definition of z and applying the approximation r5L; r5s (close to the anode wire) and sinðpr expðiWÞ=sÞ is same as ðpr expðiWÞ=sÞ in the general expression of jEj; we arrive at the expression for electric field close to the anode wire: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Q jEj ¼ 1 2 sin Wðpr=sÞ þ ðpr=sÞ2 : r From the above expression it is clear that jEj is 2Q=r if 2ðpr=sÞo1; i.e. ro0:16s:
References [1] F. Binon, et al., Nucl. Instr. and Meth. 94 (1971) 27. [2] A. Breskin, et al., Nucl. Instr. and Meth. 165 (1979) 125. [3] A. Breskin, R. Chechik, N. Zwang, IEEE Trans. Nucl. Sci. NS-27 (1980) 133. [4] A. Breskin, Nucl. Instr. and Meth. 141 (1977) 505.
[5] K. Assamagan, et al., Nucl. Instr. and Meth. A 426 (1999) 405. [6] R. Grove, et al., Nucl. Instr. and Meth. 89 (1970) 257. [7] C.J. Borkowski, M.K. Kopp, IEEE Trans. Nucl. Sci. NS19 (1972) 161. [8] J.L Alberi, V. Radeka, IEEE Trans. Nucl. Sci. NS-23 (1976) 251. [9] V. Radeka, R.A. Boie, IEEE Trans. Nucl. Sci. NS-27 (1980) 351. [10] V. Radeka, IEEE Trans. Nucl. Sci. NS-21 (1974) 51. [11] Y. Eyal, H. Stelzer, Nucl. Instr. and Meth. 155 (1978) 157. [12] J.L.C. Ford, Nucl. Instr. and Meth. 162 (1979) 277. [13] G.W. Fraser, et al., Nucl. Instr. and Meth. 180 (1981) 269. [14] H. Fanet, J.C. Lugol, Nucl. Instr. and Meth. A 301 (1991) 295. [15] G.F. Knoll, Radiation Detection and Measurement, 2nd Edition, Wiley, New York. [16] I. Endo, et al., Nucl. Instr. and Meth. 188 (1981) 51. [17] Draper, Applied Regression Analysis, Wiley, New York, 1981. [18] G.A. Erskine, Nucl. Instr. and Meth. 105 (1972) 565. [19] P.M. Morse, H. Feshbach, Methods of Theoretical Physics, Part 2, McGraw-Hill, New York, 1953, pp. 1235–1237.
Performance of a position sensitive low-pressure wire chamber (LPWC) having position readout from a separate sense wire plane: a critical analysis Chinmay Basu*, B.P. Das, Subinit Roy, P. Basu, H. Majumdar, M.L. Chatterjee Saha Institute of Nuclear Physics, 1/AF, Bidhan Nagar, Kolkata 700064, India Received 19 June 2000; received in revised form 26 July 2001; accepted 17 September 2001
Abstract The paper describes the operation of a simple one-dimensional LPWC from a completely different standpoint. The position output is derived from a separate position plane, where discrete resistances are soldered between two adjacent wires. Theoretical formulations for the position extraction are derived where the role of various detector parameters are dealt with. Position spectrum is shown for various choices of resistances varying from 50 O to 220 kO: Best position resolution and linearity is obtained for 33 kO though it violates the existing hypothesis. r 2001 Elsevier Science B.V. All rights reserved. PACS: 29.40.Cs; 29.40.Gx Keywords: Wire chamber; Position-sensitive; Resistive charge-division
1. Introduction Low-pressure position sensitive multiwire proportional counters (LPMWPC) have been widely used to detect heavy ions for their excellent position resolution and ultrafast timing properties [1–4]. A brief review of these types of detectors is given in Ref. [5]. The LPMWPC is a proportional counter with three electrodes viz., an anode wire plane placed between two cathode planes parallel to one another. MWPCs operating at relatively *Corresponding author. Tel.: 091-033-3374321; fax: 091-0333374637. E-mail addresses: [email protected] (C. Basu), [email protected] (P. Basu).
higher gas pressures (> 10 Torr) were never considered as time measuring devices because their time resolutions are typically > 50 ns: This is due to the long drift time of the electrons, released in the sensitive volume by the penetrating ionizing radiation, in reaching the anode wire, where amplification occurs. However the amplification mechanism for low-pressure counters is entirely different. The reduced electric field strength E=p (E ¼ potential difference per unit length in the sensitive volume and p ¼ pressure of the gas within the detector volume), in the constant field region, reaches a value of the order of several hundred V cm1 Torr1 : Primary electrons are accelerated in this strong uniform electric field between the electrodes, which causes ionization of
0168-9002/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 2 0 3 4 - 4
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C. Basu et al. / Nuclear Instruments and Methods in Physics Research A 484 (2002) 407–418
additional atoms. Successive ionization will result in the formation of the so-called Townsend avalanche. A primary electron created near the cathode is multiplied by several thousand before it reaches the vicinity of the anode wire. Close to the wire, the electric field strength increases even more strongly (D1=r) leading to a second electron multiplication in its vicinity. The total gain of such LPMWPC, due to this two-stage amplification process is typically 106 –107 : Thus output pulses have large amplitude, especially for heavy charged particles. Due to this strong electric field the drift time of these electrons are reduced considerably and one easily obtains rise times of the order of 1–5 ns: Thus, a time resolution of several hundred picoseconds (FWHM) can be reached under these conditions, depending on the size and geometry of the device. As the spatial distribution of each avalanche is localized around each anode wire (within 71–2 times the diameter of the wire), a method to derive the position and/ or angular information of the incident charged particle can be easily realized. Several methods of extracting the position information have been described in literature viz., center of gravity method, delay line technique, charge division method, etc. In the ‘delay line’ method, fixed delays (B2–5 ns) are introduced between each successive cathode or sense-wire plane. If the induced pulses are viewed from one end, then depending on the point of impact it will see different delays. This is an obvious approach because of the accuracy and simplicity of this type of readout methods [6]. The second method of position determination utilizes the resistive electrodes. This might be the resistive anode plane or, a separate position readout plane between the anode and cathode. The distributed resistance of this plane and the distributed capacitance between the anode and cathode make the LPMWPC equivalent to a distributed RC line. Radiation striking the detector injects a charge impulse at the point of incidence and both the amplitude and rise time of the current pulses observed at each end of the counter are dependent on the position of incidence. Thus in this method, again there are two ways of position determination, viz., the ‘‘rise time method’’ and the ‘‘charge division method’’. An
elaborate discussion about these methods can be found in Refs. [7,8], respectively. In the charge division method, position is determined by the ratio of the charge flowing out at one end of the resistive electrode, terminated into low impedance, and the total charge. The position information can be obtained either, by the evaluation of the centroid of this charge distribution [9], or a relatively simpler method is to use a global readout technique [10]. The latter method measures the charge output at the two ends of this resistive electrode. In the present work, we use a simple variant of the classical LPMWPC; with only one cathode plane instead of two. Both the cathode and anode are in the form of thin wire planes. The electric field distribution for this structure is similar to a MWPC and the mode of operation involves twostage amplification as described for a LPMWPC. However, owing to its difference in design from a classical MWPC we shall refer the present detector in this article as a low-pressure wire chamber (LPWC). In order to obtain position information, an additional wire plane is inserted between the anode and cathode. Eyal and Stelzer used a similar structure in the past for a two-dimensional PPAC [11]. The method of resistive charge division has been adopted in this work for determination of position. We have tried to characterize the LPWC with discrete resistance readout, using the global readout technique. This technique reduces complications in the readout electronics and proves to be more cost effective. This is more important in the present context with the rapid development and usage of large area detectors and multi-detector arrays. Moreover, we use the charge dividing resistors on the position plane, instead of the more widely used resistive anode charge division [12] or cathode strip readout [9]. Though earlier workers [8,10,13,14] have performed extensive studies on RC encoding, the discussions were confined more towards optimization with regard to linearity and resolution, in terms of theoretical noise analysis and optimum filtering. The effects of practical deviations occurring in a real (operating) position sensitive detector, from the commonly assumed point injection of charge has been less considered.
C. Basu et al. / Nuclear Instruments and Methods in Physics Research A 484 (2002) 407–418
In this work, we study resistive charge division by two-channel readout with equal emphasis on theory and experiment. An elaborate discussion into the theoretical aspects of the LPWC has been included wherein the role of various parameters have been critically examined. Optimization of various parameters like resistors, pressure of the gas, reduced electric field and shaping time of the amplifier have been carried out in conjunction with the theoretical interpretations. A position resolution of less than a millimeter has been obtained inspite of the crudity of the global readout method. The paper is organized as follows. In Section 2 we describe the construction of the detector. The consistency of the detector behavior has been analyzed with the existing theoretical in terpretations in Section 3. Experimental details and discussion of the results are given in Section 4. Finally, summary and conclusions are given in Section 5.
2. Construction of the detector The constructional details of the counter are shown in Fig. 1. The cathode and anode wire planes are made of 12:5 mm thick gold-plated tungsten wires soldered on a copper clad G-10 board at a spacing of 1 mm: Use of thin wires in the cathode plane is a source of field emission. This problem is averted by the use of thin conducting foil [3]. Our primary objective however, of fabricating the present detector, is to use it in the focal plane of a magnetic spectrograph for efficient measurement of low energy, high Z evaporation residues. Use of metallized foils would reduce the
409
transmission drastically. To extract position information, a separate plane is inserted (Fig. 1) between the anode and cathode, with wires thin enough, to minimize distortion of field lines. The sense wires are 25 mm in diameter and spaced 2:5 mm apart. Discrete resistors are soldered between each adjacent sense wire. The value of the resistors used in our experiment varied between 50 O and 220 kO: Two such prototype detectors, having areas 90 65 and 75 50 mm2 were used. The distance of separation between cathode and anode wire planes were varied between 7 and 10 mm; whilst that between sense wire and anode is kept fixed at 3 mm: The gas windows are made of 50 mg cm2 stretched polypropylene film. A primary objective of using large area position sensitive gas counters is to utilize its very high rate handling capacity (B106 counts s1 ). However, charge division method is inherently slow compared to other methods of position determination, viz., the delay line or the center of gravity method. A possible way of eliminating this problem is to place the position plane close enough to the anode for fast and efficient position determination. Normally, one uses the cathode plane to extract position information. It is our experience, however, that there is always a problem of sparking when the anode–cathode gap is made small and stable operation of the counter for extended periods become impossible. Besides, the separate sense plane becomes useful in obtaining twodimensional readout if the cathode is also used as a position plane [11]. The detector was operated with 99% pure isobutane gas at a pressure varying between 2– 5 Torr: The pressure inside the detector volume
Fig. 1. The schematic diagram of the low pressure position sensitive wire counter (LPWC). The anode A (12:5 mm gold-plated tungsten wire, 1 mm apart), the cathode C (12:5 mm gold-plated tungsten wire, 1 mm apart) and the sense wire plane S (25 mm gold-plated tungsten wire, 2:5 mm apart) are shown. The wires are soldered on G-10 boards. The gas window and supporting mesh are not shown. Typical voltages applied on the anode and cathode wire planes are also shown.
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was kept fixed during the experiment by means of an electronic pressure-regulating valve. A well-collimated 252 Cf fission source (strength 10 mCi) was used during testing of the LPWC. The source to detector distance was kept at 10 cms: The typical anode pulse characteristics for our detector and also for an equivalent LPMWPC are given in Table 1. In spite of the simple design of the present chamber, the pulse height and rise times are of the same order as in a LPMWPC. The schematic layout of the electronics is shown in Fig. 2. The charge signals from the two ends of Table 1 Typical anode pulse characteristics of fission fragments detected from a 252 Cf source with the detector described in this work and a LPMWPCa Anode pulse characteristics
Anode–cathode gap (mm)
LPWC (present detector)
LPMWPC
Pulse height
4.0 8.0
500 mV 700 mV
1V 1:5 V
Rise time
4.0 8.0
4 ns 7 ns
4 ns 7 ns
a The operating pressure was kept at 3 Torr with E=P around 250 V cm1 Torr1 : The pulse amplitude and risetimes obtained after a VT120, Gain 200, ORTEC fast preamplifier and recorded in a 500 MHz HP Oscilloscope are depicted in the table. The noise level and the approximate gain of the present detector are also shown. Noise ¼ 15–20 mV; Gain (LPWC)B1056 :
Fig. 2. Simplified detector and electronic circuit for position measurements with the LPWC using the charge division technique.
the sense wire planes were taken through conventional charge sensitive preamplifiers (ORTEC 142IH). They were subsequently shaped and amplified through active filter amplifiers (ORTEC 572) and summed to give the total charge. The ratio of one end of the signal to the total gives the position information. Before the experiment, the preamplifiers and amplifiers were properly gain matched by a research pulser. The anode and cathode volt ages were given through ORTEC bias supplies.
3. Mechanism of charge division in the low-pressure wire chamber The principles regarding the operation and characteristics of the basic proportional counter and the development of pulses within these detectors have been extensively reviewed [15]. However, essential features of the theory of operation of the present counter as a position sensitive device will now be discussed in order to provide a physical understanding of their properties. The mode of operation of the present lowpressure wire chamber (LPWC) is similar to that of a conventional low-pressure MWPC. In the LPWC, a two-stage amplification of the electrons take place, which essentially depends on the reduced electric field E=p and the gas used. The electric field is constant in the drift region (at distances greater than several times the anode wire radii). However, the value of E=p here is of the order of several hundred V cm1 Torr1 ; and is sufficient for secondary ionization. Nearer to the anode, the field varies as 1=r- (see appendix) and a second stage of amplification sets in. This latter amplification of electrons occurs very close to the anode wire and is localized in space. They give rise to an ultrafast anode signal (Bns). The purpose of the position readout is to sense these avalanches. The avalanches created near the anode will induce a charge signal on the nearby sense wire. The induced charge will flow out at the two ends of the position readout plane with a magnitude dependent on the total resistance faced in its path. If we consider this resistive electrode as an effective RC transmission line, with distributed R and C per
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411
unit length, the charge division and diffusion in one dimension can be treated through Telegrapher’s equation [12] (with G (conductance) and L (inductance) being negligible):
Considering a point injection of charge Q at x ¼ x0 (avalanche point) and at time t ¼ 0; we have in terms of the Dirac d-function, dðx x0 Þ;
@2 uðx; tÞ @uðx; tÞ ¼ RC @x2 @t
uðx0 ; 0Þ ¼
ð1Þ
where uðx; tÞ is the voltage induced on the line at position x and time t; R the resistance per unit length and C the capacitance per unit length. Eq. (1) is a standard diffusion equation in onedimension. It can be solved by means of separation of variables X uðx; tÞ ¼ Xn ðxÞTn ðtÞ: ð2Þ
Q dðx x0 Þ: C
ð8Þ
Substituting Eq. (8) in Eq. (7) and using Fourierseries expansion, we finally arrive at N npx npx 2Q X 2 2 2 0 uðx; tÞ ¼ sin eðn p t=RCl Þ : sin Cl n¼1 l l ð9Þ
Substitution of Eq. (2) into Eq. (1) gives
In order to arrive at the expression for current from Eq. (9), we use the first coupled Telegrapher’s equation
1 d2 Xn ðxÞ RC dTn ðtÞ ¼ l2n ðconstantÞ: ¼ Xn ðxÞ dx2 Tn ðtÞ dt
@iðx; tÞ @uðx; tÞ þC ¼0 @x @t
n
ð3Þ Solving for the time and position part separately gives 2
Tn ðtÞ ¼ Tn ð0Þeðln t=RCÞ
wherefrom iðx; tÞ ¼
npx npx 2pQ X 0 n sin cos RCl 2 n l l
ð4Þ 2 2
eðn p
and, Xn ðxÞ ¼ An sinðln xÞ:
ð5Þ
For the latter, we have assumed both ends of the line being terminated by zero impedance preamplifier, i.e. uðx ¼ 0; tÞ ¼ uðx ¼ l; tÞ ¼ 0; where l is the length of the line. From Eq. (5) this gives np ln ¼ : ð6Þ l Therefore, we finally arrive at N npx X 2 2 2 uðx; tÞ ¼ Cn sin eðn p t=RCl Þ ; l n¼1
ð10Þ
ð7Þ
where An and Tn ð0Þ are absorbed in Cn : The above expression signifies the voltage distribution along a one-dimensional distributed RC-line, for any point x along the length of the line and at any finite time t: The solution consists of a sinusoidal space part and an exponentially decaying time part, both of which depends on the line parameters, viz., R; C and l:
t=RCl 2 Þ
:
ð11Þ
From the above expression the values of the current, i; at the two ends, i.e., x ¼ 0 and l will be npx 2pQ X 2 2 2 0 n sin ið0; tÞ ¼ eðn p t=RCl Þ 2 RCl n l and iðl; tÞ ¼
npx 2pQ X 2 2 2 0 n sin cosðnpÞeðn p t=RCl Þ : 2 RCl n l
Since Qð0Þ þ QðlÞ ¼ Q; is a conserved quantity, we define a convention that Qð0Þ decreases with increasing x and QðlÞ increases with increasing x: Therefore, we arrive at Z t ið0; t0 Þ dt0 Qð0; tÞ ¼ 0
¼
2Q X sinðnpx0 =lÞ 2 2 2 ð1 eðn p t=RCl Þ Þ p n n ð12Þ
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and Qðl; tÞ ¼
Z
t
iðl; t0 Þ dt0 0
¼
2Q X sinðnpx0 =lÞ cosðnpÞ p n n 2 2
ð1 eðn p
t=RCl 2 Þ
Þ:
ð13Þ
The time varying part of the charge expressions (12) and (13) are averaged over a time equal to the amplifier shaping time (tF ) by the appropriate weighting function of the amplifier. Alberi and Radeka [8,10] carried out similar averaging, though for a trapezoidal weighting function. In our case we have used ORTEC 572 amplifiers, which uses an approximation to Gaussian shaping. Incorporating this weighting function into the above equation and simplifying we finally obtain x0 px0 Q1 ¼ Qð0; tF ÞDQ 1 ð14Þ QK sin l l Q2 ¼ Qðl; tF ÞDQ
x0 px0 QK sin l l
ð15Þ
px0 Q1 þ Q2 ¼ Q 2QK sin ð16Þ l where K is the non-linearity term which for a semiGaussian weighting function is obtained as eð1XÞ 2 2e 1 X þ K¼ ð17Þ 1 X X2 X X where X ¼ ðp2 tF =tD þ 1Þ; tD being time constant of the line, i.e. tD ¼ RCl 2 : Thus, using a finite integration time, a nonlinearity K is introduced which is a function of tF =tD : Ideally, for integration time tF -N; K-0; and the asymptotic charge expression at the two ends of the line becomes 2Q X sinðnpx0 =lÞ x0 ¼Q 1 Q1 ¼ Qð0Þ ¼ ; p n n l ð18Þ where the final equality results from the standard summation of the series. Similarly from Eq. (15), using tF -N; we arrive at the expression for the other end Qx0 Q2 ¼ QðlÞ ¼ : ð19Þ l
The position signal is obtained by taking the ratio of charge at one end to the total charge, i.e., Q1 =Q1 þ Q2 or Q2 =Q1 þ Q2 : It is easily seen from Eqs. (18) and (19) that position signals are strictly proportional to x0 =l (distance of avalanche from one end). However, one must keep in mind that we have taken the asymptotic charge expression (tF -N). This is an ideal situation not realized in practice. Let us now consider how the situation modifies for finite shaping times (Bms for spectroscopy amplifiers). The position signal for finite tF can be constructed from Eqs. (15) and (16):
Q2 x0 2x0 px0 : ð20Þ E K 1 sin Q1 þ Q2 l l l Thus, there is always some non-linearity associated with finite tF ; depending on K: We now define the dynamic range of the spectrum, D as
Q1 Q1 D¼ ð21Þ Q1 þ Q2 max Q1 þ Q2 min in terms of the maximum and minimum extent of the position spectrum. Ideally, for tF -N; D ¼ 1: But for finite shaping time, we can write the dynamic range (after some algebraic manipulation of Eq. (20)) as k1 pk p 1 þ K sin þ sin DE ð22Þ kþ1 kþ1 kþ1 where k is the number of sense wires used. It can be easily seen from Eqs. (22) and (17) that the dynamic range increases with K; i.e. for larger tD (high resistance). In fact this effect may result in D exceeding unity for a large value of resistance. It can also be visualized that with fixed tD ; if we increase the shaping time, tF ; of the amplifier, the dynamic range decreases. This can lead to squeezing of the spectrum at higher shaping times. Thus, a proper choice of the value of resistance used and the shaping time of the amplifier circuit is of prime importance. In the above discussion, we have assumed a point injection of charge (Eq. (8)) at x ¼ x0 and t ¼ 0: However, for our detector configuration, we have a separate sense wire plane at a distance ‘L’ from the anode plane (position of actual avalanche) and the individual sense wires are
C. Basu et al. / Nuclear Instruments and Methods in Physics Research A 484 (2002) 407–418
separated by a distance ‘s’. This configuration will lead to a charge distribution in the sense plane, and the previous point charge hypothesis has to be modified. A charge distribution, used earlier [16] would be appropriate Q pðxi x0 Þ qðxi Þ ¼ sec h ð23Þ 4L 2L where x0 is the position of actual avalanche at anode, xi is the position of ith sense wire form one end and L the distance of separation of anode and sense wire plane. Considering a linear charge distribution one can write the ‘position’ or the charge ratio, in terms of the charge distribution as Rl qðxÞðl xÞ dx Q1 ¼ 0 Rl Q1 þ Q2 qðxÞl dx 0
Rl ¼ Z
0
qðxÞðl xÞ dx ; Q:l
ð24Þ
l
qðxÞ dx ¼ Q; 0
where qðxÞ is the linear density of charge. The above expression is valid for a continuously varying resistive plane. For a readout from a sense wire plane, as in our case, with discrete number of wires, we can modify Eq. (24) to read Pn Q1 qðxi Þðl xi Þ ; ð25Þ ¼ i¼1Pn l i¼1 qðxi Þ Q1 þ Q2 where ns ¼ l; n being the total number of sense wires. Using the charge distribution (23) for the sense wire, and applying it to Eq. (25), the behaviour of the position signal (charge ratio) on L and s can be examined. Eq. (25) essentially signifies that though the avalanche is localized on one particular anode, the charge ratio Q1 =Q1 þ Q2 is actually obtained by summing contributions from all sense wires. The magnitude of charge injected on each sense wire (a fraction of Q) gets progressively reduced as we go away from the point of localization. That is, for a sense wire plane kept at some distance away from the anode plane (plane of localization), the mechanism of charge division is different from that of point injection of charge.
413
In the preceding discussions, we have briefly reviewed the theory of charge diffusion for point charge injection. Non-linearity in the position extraction has been derived for a semi-Gaussian weighting function and is seen to depend on tF =tD : Some simple calculations were also done to treat the mechanism of charge division on a separate position plane. For our detector configuration, it is a charge distribution that one has to deal with instead of point charge injection. How these factors affect the detector resolution and linearity will be considered in the next section.
4. Results and discussions The primary motive of the experiment is to find out proper choice of resistance, (r), used between adjacent sense wires such that the position resolution and/or linearity is optimum. In order to arrive at a definite conclusion we have used various discrete r between each successive sense wire. The values of r used ranged from 50 O to 220 kO: Our detector dimension could accommodate a maximum of 30 resistors for the smaller detector (75 mm 50 mm) and 36 for the larger one (90 mm 65 mm). The capacitance of these prototype detectors were also measured and found to be B20 pF: Thus the RC-value (denoted by tD ; the time constant of the resistive circuit) of sense wire line for each r can easily be calculated and are tabulated in Table 2. We have measured the position resolution for each choice of resistance value. This was done by uniformly illuminating a mask, having 1 mm wide slits placed 1 mm apart. The pressure of the isobutane gas within the detector volume was kept fixed between 3 and 5 Torr: The reduced electric field E=p was varied between 150 and 350 V cm1 Torr1 : Bias voltages were applied on both the anode and cathode planes while keeping the sense wire plane floating. The schematic diagram of the measuring circuit is shown in Fig. 2. The charge outputs from the two ends of the RC-line were fed to a charge sensitive preamplifier (ORTEC 142IH in our case) and then shaped and amplified in a spectroscopy amplifier (ORTEC 572 in our case) which provides
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Gaussian filtering (CR-RC6 ). They are further processed in a dual sum and invert (ORTEC 533) and analog divider (ORTEC 464), properly calibrated to ensure non-linearity within 0:1%: All earlier studies on RC-encoding by charge division was made from a more theoretical viewpoint using idealization such as point charge injection. Gas counters operating at higher pressures have only been considered and ratio of filtering time constant (tF ) to the RC line time constant (tD ) was shown to be an important factor [8,10,13]. So in this work, the experiments were performed at different amplifier shaping times for each choice of r: We have shown in Table 3, values of the parameter tF =tD ; for various choices of r and available settings of amplifier shaping times ðtF Þ: The values of tF =tD are very high for lower r and decreases as we progressively increase the value of resistance. Fig. 3 shows the results of the experimental position spectrum for various r and tF : The values Table 2 Value of RC (denoted by tD ) for different combinations of resistors and detector dimensions. The capacitance of the detector measured from end to end is B20 pF r (in O)
tD (in ms)
Detector 1 ð90 mm 65 mmÞ 36 resistors
50 100 200
0.036 0.072 0.144
Detector 2 ð75 mm 50 mmÞ 30 resistors
680 4:7 103 10 103 33 103 220 103
0.4 2.82 6.00 19.80 132.00
of the factor tF =tD are also shown within parentheses for each figure. The different choices were selected to represent the best spectrum for each resistance value. If we scan through the different choices (Fig. 3(a)–(f)), it can easily be surmised that two values r ¼ 680 O (Fig. 3(b)) and r ¼ 33 kO (Fig. 3(e)) are the better choices. Clearly separated peaks and excellent position resolution (o1 mm FWHM) are achieved. The peaks also rise from the baseline showing negligible component of background noise. For the 33 kO case the intensity distribution is also in consonance with placement of a well-collimated source at the mid-point of the detector area. For other choices of r; some proportion of noise is always present. In Table 4, we have given for some representative cases the position resolution (FWHM) with reference to 1 mm slit and the limit to resolution in each case due to electronic noise. To determine the latter, equal charge from a pulser were fed to the two preamplifier test inputs at the two ends of the RCline which was kept floating. The position signal FWHM obtained is indicated in units of mm. One interesting aspect of the experiment is that in general we get fairly good resolution (o1 mm FWHM) over a wide range of tF =tD (see Table 4), whereas earlier workers [8,10,13] concluded that good resolution is expected only when tF =tD 51: For r ¼ 100 O (Fig. 3(a)) and other low values of r below 680 O there is significant amount of noise for all values of amplifier shaping times. However, it appears that the position resolution is better at lower values of tF =tD (Table 4), similar to the trend predicted by Refs. [8,10,13]. Fig. 4 is a representation of the position linearity. Shown here are the distance of the
Table 3 Value of the parameter tF =tD for different choices of resistors used and also for available settings of amplifier time constants (tF ) Shaping time tF ðmsÞ 0.5 1.0 2.0 3.0 6.0 10.0
tF =tD 50 O
100 O
200 O
680 O
4:7 kO
10 kO
33 kO
220 kO
13.9 27.8 55.6 83.4 166.8 278.0
7.0 13.9 27.8 41.7 83.4 139.0
3.5 6.95 13.9 20.85 41.7 69.5
1.22 2.44 4.88 7.32 14.64 24.40
0.175 0.348 0.695 1.043 2.085 3.475
0.083 0.167 0.333 0.501 1.002 1.67
0.025 0.050 0.101 0.151 0.303 0.505
0.0038 0.0080 0.0150 0.0230 0.0450 0.0760
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(a)
600
τF=0.5 µs r =100 Ω
τF=0.5 µs, r =680 Ω
80
(7.0)
400
(b)
100
(1.22)
60 40
200
20 0
0 250
750 1000
0
250 500 750 1000
(c)
400
Counts
500
300
(d)
50
τF=2 µs r =4.7 kΩ (.695)
τF=3 µs r =10kΩ (.501)
40 30
200
20 100
10
0
centers of various slits (uniformly spaced) against each peak channel number (y-axes). Ideally for precise position information it should strictly fall on a straight-line fit (linear regression [17]). The goodness of fit may be quantified by the correlation coefficient c; of the regression line. This is indicated in Figs. 4(a)–(f) within brackets. Closer the value of c to unity, better is the fit and hence the linearity. However, we see that there are small oscillations about the mean straight line. These minor variations represent the differential nonlinearity associated with each setting. A striking feature of the non-linearity is that they are oscillatory in nature. This is understood from the
0 250
500
750 1000
(e) τF=1 µs r =33 kΩ
100
(.05)
80
600
250 500 750 1000
(f)
120
1000 800
0
(a) 2500
τF=10 µs r =220 kΩ (.076) (.076)
2000
200 0
40
1000
20
500
250 500 750 1000
220kΩ,τF=6 µs
τF/τD=0.0075
1500
τF/τD=0.045
1000 500 (0.993)
(0.971)
0 0
220kΩ,τF=1 µs
1500
60 400
(b) 2000
0
0 0
250 500 750 1000
0 10 20 30 40 50 60
0 10 20 30 40 50 60
Channle
(c) 1200
Peak Channel
Fig. 3. (a)–(f) Position spectrum obtained for various combinations of tF and r: The numbers within the brackets are the ratio tF =tD : The spectrum is obtained by uniformly illuminating a rectangular mask having 1 mm slits and 1 mm opaque region.
1000 800
(d )
33kΩ,τF=1 µs
2000
τF/τD=0.05
1600
680Ω,τF=0.5 µs τF/τD=1.22
1200
600
800
400 200
(0.997)
0
400
(0.999)
0 0 10 20 30 40 50 60
0 10 20 30 40 50 60
(f)
(e) Table 4 Experimental position resolution (FWHM) expressed in millimeter for different choices of tF =tD . Value of noise (FWHM) is also indicated in millimeter tF =tD 0.050 0.076 0.083 0.150 0.500 1.220 7.000 7.320 24.400
r ðOÞ 33 k 220 k 10 k 33 k 33 k 680 100 680 680
tF ðmsÞ 1.0 10.0 0.5 3.0 10.0 0.5 0.5 3.0 10.0
FWHM (mm) 0.757 0.779 0.797 0.857 0.925 0.748 1.75 0.842 0.902
Noise FWHM (mm) 0.360 0.104 0.070 0.180 0.230 0.430 0.940 0.150 0.180
1200
1800 1600
680Ω,τF=10 µs
1400
τF/τD=24.4
1000
100 Ω, τF=0.5,2 µs
800
1200
600
1000
400
800
200
(0.999)
τF/τD=7.0 τF/τD=27.8 (0.993,0.987)
0
600 0
10 20 30 40 50
0 10 20 30 40 50 60 70
Position (mm) Fig. 4. (a)–(f) Linearity plots (peak channel vs. position) obtained for various choices of tF and r: Values of tF =tD are also indicated. The experimental points are indicated by dots. The straight-line fits (linear regression) are also shown. The correlation coefficients of the regression line are indicated within brackets.
C. Basu et al. / Nuclear Instruments and Methods in Physics Research A 484 (2002) 407–418
position term in Eq. (20) for finite tF ; derived for point charge injection. The oscillations should have intercepts at 0, l=2 and l on the straight line. In our case the source has an extended charge distribution at the position plane and so the oscillations show additional intercepts in some cases (Fig. 4(b)). In Section 3, assuming point charge injection, we derived the non-linearity factor K for the Gaussian filter (Eq. (17)) which was found to be decreasing with increasing values of tF =tD : This trend is also reproduced in our experimental observation in Fig. 4, except for r ¼ 100 O (Fig. 4(f)). However, we get acceptable experimental linearity (c ¼ 0:993; 0:997) even when tF is only 5% of tD (Fig. 4(b) and (c)) where K from Eq. (17) has a large value of about 0.7 (linearity 0.3). This indicates that K needs to be redefined for extended charge distribution. To make a more stringent choice of r; we further compare in Fig. 5 the position spectrum for 680 O and 33 kO; at three different shaping times. It is clearly seen that the dynamic range of the position spectrum gets progressively reduced for higher tF ; predominantly for r ¼ 680 O: The reduction for the other resistance value r ¼ 33 kO is minimal. At the highest shaping time (6 ms), the range for 680 O is already 30% reduced compared to 33 kO: Therefore, we make two significant observations: (a) For a particular choice of r; the dynamic range gets smaller with increasing tF and (b) The squeezing effect is more pronounced for smaller r: Effect (a) can be understood with reference to Eq. (17), where for a particular tD with increasing tF ; K decreases, thereby decreasing D: Effect (b) is a result of faster decrease of K with the increase of tF =tD for 680 O; in comparison to r ¼ 33 kO:
5. Summary and conclusions The paper describes the operation of a simple low-pressure wire chamber from a different stand-
τF/τD 600
τF=1µs
r =33 kΩ
0.05
300 100 r =680 Ω
60
τF=1µs
40
2.44
20 0 0
500
600
Counts
416
1000
1500
2000 τF=3µs
r =33 kΩ
0.151
300 100 60
τF=3µs
r =680 Ω
40
7.32
20 0 0
500
1000
1500
2000 τF=6µs
r =33 kΩ
150
0.303 50 60
=6µs ττFF=6µs
r =680 Ω
40
14.64
20 0 0
500
1000
1500
2000
Channel Fig. 5. Comparison of position spectrum for two choice of resistance values, r ¼ 680 O and 33 kO; shown for different shaping times (tF ¼ 1; 3; 6 ms). The value of the ratio tF =tD is given for each spectrum.
point. In our detector configuration, the position grid is a separate wire plane with discrete values of resistors between two adjacent wires. Position resolution and linearity have been measured over a wide range of resistance values (50 O–220 kO). The role of the finite shaping time of the amplifier in the proper choice of resistance value has been considered. Finally, theoretical insights into the dependence of various parameters, both for the ideal case (tF -N) and real situation (tF finiteEms) are also provided. Theoretical aspects of charge division on a separate wire plane
C. Basu et al. / Nuclear Instruments and Methods in Physics Research A 484 (2002) 407–418
considering spatially extended charge distribution has also been discussed. The conclusions can be classified broadly for (a) Lower values of resistors: The RC-line noise becomes an important criterion. This worsens the position resolution. The dynamic range becomes small when r is less than or close to the preamplifier input impedance. Linearity is affected. A strong squeezing effect is also observed in the position spectrum with increasing amplifier shaping time. (b) Higher value of resistors: The line noise and squeezing effect gets considerably reduced. Measured position resolution is less than a millimeter over a wide range of amplifier shaping times. The observed position linearity is acceptable even at a low value of the ratio of the filtering time constant to the RC-line time constant. A deviation from the criterion for optimum linearity and resolution (tF =tD > 0:5) deduced earlier [8,10] is thus observed. To account for this discrepancy, redefinition of the theoretical non-linearity considering (a) extended charge distribution and (b) pure Gaussian filter is suggested. A detailed theoretical calculation will be extremely complicated and may be pursued in future. We finally conclude that higher resistance values are preferable and consider r ¼ 33 kO to be the optimum choice for our detector design. The present study provides an extension of earlier results [8,10,13,14] on RC encoding. A specific analysis has been carried out that has not been probably done before, viz. the response of a RC electrode to a spatially extended charge injection in a low-pressure wire chamber.
Acknowledgements The authors thankfully acknowledge the kind interest of Professor Sudip Ghosh in the present work and the support and encouragement received from him.
417
Appendix A The electric field distribution for a MWPC of classical design with two cathodes and the anode in between is well known [18,19]. The detector in the present work has only one cathode placed parallel to the anode (Fig. 1). The basic electric field distribution is however similar to that of a standard MWPC. The potential at any point z in the complex plane due to a plane of uniformly charged parallel thin wires in front of a conducting plane cathode (an equivalent image structure is that of the present detector structure) is given by Ref. [19] as FðzÞ ¼ 2Q ln
sin pðz iLÞ=s ; sin pðz þ iLÞ=s
where Q is the charge per unit length and is given as V0 =f2pðL=sÞ 2lnðpd=sÞg; in terms of the anode potential V0 ; anode–cathode gap L; wire diameter d and inter wire separation s: The general expression for electric field is given by
@
jEj ¼
FðzÞ
@z
2pQ
pðz iLÞ pðz þ iLÞ
¼ cot cot
: s s s A.1. Field away from the anode wire With the origin of the z plane on the cathode, this region is where jzj5L: The electric field can then be written as
2pQ
pðiLÞ pðiLÞ
jEjE cot cot s s s 4pQ pL 4pQ coth E : s s s The last expression can be written, as cothðpL=sÞ tends to 1 as pL=sb1: Thus, the electric field is constant away from the anode wire. ¼
A.2. Field close to the anode wire In this region it is convenient to use the ðr; WÞ definition of z as in Ref. [19], i.e.,
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C. Basu et al. / Nuclear Instruments and Methods in Physics Research A 484 (2002) 407–418
z ¼ iL þ r expðiWÞ; where r is measured from the wire as origin. Using this definition of z and applying the approximation r5L; r5s (close to the anode wire) and sinðpr expðiWÞ=sÞ is same as ðpr expðiWÞ=sÞ in the general expression of jEj; we arrive at the expression for electric field close to the anode wire: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Q jEj ¼ 1 2 sin Wðpr=sÞ þ ðpr=sÞ2 : r From the above expression it is clear that jEj is 2Q=r if 2ðpr=sÞo1; i.e. ro0:16s:
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