Analysis of the position resolution in centroid measurements in MWPC

Nuclear Instruments and Methods 188 (1981) 327-346 North-Holland Publishing Company

327

A N A L Y S I S O F T H E P O S I T I O N R E S O L U T I O N IN C E N T R O I D M E A S U R E M E N T S IN

MWPC

*

E. G A T T I , A. L O N G O N I Istitutto di Fisica del Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

R.A. B O l E and V. R A D E K A Brookhaven National Laboratory, New York 11973, USA

Received 13 March 1981

Resolution limits in avalanche localization along the anode wires of an MWPC with cathodes connected by resistors and equally spaced amplifiers, are evaluated. A simple weighted-centroid method and a highly linear method based on a linear centroid finding filter, are considered. The contributions to the variance of the estimator of the avalanche position, due to the series noise of the amplifiers and to thc thermal noise of the resistive line are separately calculated and compared. A comparison is made with the resolution of a MWPC with isolated cathodes. The calculations are performed with a distributed model of the diffusive line formed by the cathodes and the resistors. A comparison is also made with the results obtained with a simple lumped model of the diffusive line. A number of graphs useful in determining the best parameters of a MWPC, with a specified position and time resolution, are given. It has been found that, for short resolution times, an MWPC with cathodes connected by resistors presents better resolution (lower variance of the estimator of the avalanche position) than an MWPC with isolated cathodes. Conversely, for long resolution times, the variance of the estimator of the avalanche position is lower in an MWPC with isolated cathodes. 1. Introduction

The system considered is an MWPC with strip or wire cathodes c o n n e c t e d by resistors, for avalanche localization along the anode wires (figs. 1 a and b). Equally spaced charge-amplifiers followed by shaping filters are c o n n e c t e d at taps o f the diffusive line formed by the cathodes and the resistors. The centroid o f the induced charge is d e t e r m i n e d by one o f the appropriate centroid m e t h o d s applied to the output pulses of the amplifier-filter units. The purpose o f this w o r k is to find the resolution linrits, in determining the avalanche position, due to noise f r o m the amplifiers and the thermal noise o f the resistive line. A comparison is m a d e w i t h the resolution limits o f an MWPC in which the cathode strips are isolated f r o m each o t h e r (figs. l a and c) and each cathode is c o n n e c t e d to an amplifier [1]. The com-

parison holds also for an MWPC with an interwoven structure o f grid wires for the second coordinate readout [2]. A listing o f the symbols used in the present w o r k is given in A p p e n d i x 1.

Mwec DIMENSIONS

/~

0 0 2 9 - 5 5 4 X / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 5 0 © 1981 North-Holland

ELEMENTARYCELL DIMENSIONS

~,,

I-t

2a

"i

T...T_.TF..-"r~...T

-~/ ' 0~r_,/~/~a

"

~

(b~

)

EATHODE PLANE

/

2a

I°

ANODE PLANE

-F_ /I____.~ ,1 / \ _d_./ \ IJR I T~ • hllt Te

Work donc with partial support of BNL Contract DE-AC02-76CA0016 and INFN (Istituto Nazionalc di Fisica Nuclearc Italy).

L

LI

~ !

'

T__y

(c)

" Cd)

Fig. 1. (a) MWPC geometry; (b) "cell" formed by several cathodes and resistor for charge division readout; (c) "cell" for readout through charge induced at every cathode. (d) 6-current response of the amplifier-filter unit.

328

E. Gatti et al. / CentroM measurements in MWPC

2. The estimator of the avalanche position and its variance In the case o f sensing cathodes connected by resistors we must take into account the different time delay of the charge-signal at different amplifier inputs, due to the transit time along the diffusive line. To this aim, we consider in this work a trapezoidal 6-current response (fig. l d) of the amplifier-filter unit, with a fiat top of suitable duration TI.. , as suggested by Alberi and Radeka [3]. In the case of isolated cathodes directly connected to the amplifiers, a triangular ¤t response will be considered, that is the trapezoidal response in the limit T v = 0. In fact, in this case, the induced signal appears at the same instant at all taps. In practice, a short non-zero T v is always adopted for instrumental reasons (amplitude sampling). The estimator S o f the avalanche-position can be assumed to be a weighted-centroid of the peak amplitudes V m o f the signals at the output of the amplifier-filter units. ~ S

-

mPm Vm

(1)

~

Pm Vm m

In the above expression, the unit length is the distance between two contiguous amplifier taps. The integer m means the ruth amplifier and its coordinate. The weights Pm are introduced in order to restrict the number of anrplifiers for the centroid evaluation. In the simplest case, P m = 1 for the considered taps and Pm = 0 for the others. From eq. (1), the variance of the estimator S can be written as (see Appendix 2): (m

S ) ( k - S) PmPk dVm dVk

mk

0

a5

I I

T75 ~0

ss

"-Q

SEQUENTIAL

~

/

(a) I

ss % (c)

i

to /H = b

-to.POSITION ~-

I"ig. 2. (a) Block scheme of the Radeka and Boie system; (b) centroid filter a-response; (c) convolution of the waveform (b) with a rectangular pulse of width TS. can be written as explicit functions P(m S) of the avalanche position. The 6-response of the linear centroid finding filter that we have considered in this work is plotted in fig. 2b, and in fig. 2c is tire weighting function P(m .- S), resulting from the convolution of the 6-response of the filter with a rectangular pulse, representing tire output of the sample and hold circuit. The estinrator S, in this case, is implicitly defined by (see Appendix 2):

F, Vm?(m-s)

= 0.

(3)

From eq. (3), tire variance of the estimator can be calculated as: P(m-S) ff~ =

P ( k - S) dVmdVk

ttl k

v,,,~ ?(m ---s)) where d V m and dV k are the noise fluctuations of the peak values Vm and Vk and dVmdV ~ is the average product. Centroid estimators with high differential linearity can be obtained by using more elaborate weighting techniques. In this work, for instance, we consider a technique developed by Radeka and Boie [4], in which the centroid of the charge is deternfined by suitably filtering a pulse built by sequentially san> pling all the peak amplitudes V m (fig. 2). Tile chargecentroid is determined by detecting the zero crossing time of tire filtered pulse. In this case the weights Pm

Ts

(b)

(4)

3. Evaluation of the noise terms in the variance In order to evaluate tire variance e2S we first determine the "noise term", that is the term: (m- S)(k m

S ) P m t ' k dVm dVk ,

(5)

k

of eq. (2) and tile term: P(m - S ) P(k - S ) d V m dVk m k

of eq. (4).

(6)

329

E. Gatti et al. / CentroM measurements in MWPC

In the next section we will evaluate the "signal term", that is the denominators of eqs. (2) and (4). In order to determine the fluctuation dVm at the output of the ruth preamplifier-filter block the series input noise E m o f the ruth preamplifier must be considered. Moreover, in the case of the diffusive line, the effects of the noise injected by the contiguous amplifiers as well as the thermal noise of the line itself must also be taken into account. (In this work we will not consider the noise injection due to the capacity between contiguous electrodes in the case of contiguous insulated cathodes. The corresponding effects can be approximately taken into account by adopting in the calculations a suitably higher value for Era, as was done in ref. 1.) The sums (5) and (6) are conveniently evaluated by first considering as active a single amplifier noise generator E,n o f the ruth amplifier) and then superposing the uncorrelated effects due to the other noise generators. Similarly the thermal noise contribution of the line is evaluated by considering the noise generated by only a single cell o f the line [e.g. the line between the ruth and the (m + 1)th tapsl. See Appendices 3 and 4. The contribution o f a single noise source can then be split into several terms which have simple physical meaning. These terms are: a~m : this term takes into account that part o f the effects of generator E m upon the output of the ruth ampfifier due to the presence of the input capacitance C A of the amplifier. (CA = FET input + feedback capacity.) a z m : this term takes into account that part o f the effects of the generator E m upon the output of the ruth amplifier due to the impedance of the connected lines. a3m: term due to the correlation between the two preceding contributions arm and a 2 m . a4rn: this term takes into account the noise contribution of contiguous anrplifiers m - 1 and m + 1 due to the current noise that the generator E m injects into their virtual grounds through the diffusive line. asm: term due to the correlation between the

term a i m and

m÷l Im.m+l

RD m

j_ ti o [m-l.rn ~

ER°

Q_

II

IE ~Ci

m-1 Fig. 3. Lumped model for calculating the contributions of series noise generator E m and parallel current generator Ira_l, m

and Ira, m + 1 to centroid variance.

noise signals, at the outputs of the ruth and (in + 1)th amplifiers, due to the thermal noise of the line between them. The sums A1 = ~,alm, A 2 = Z a z m t e A 6 = Y-,a6m take into account the noise of all amplifiers, and, similarly, the sums A7 = £ a v m and A8 = £ a a m that of the complete line. The expressions for the terms A~, . . . , A s can be found in table 1. In this table two sets of terms appear. The one labelled as the "lumped model" represents the result of a rough approximation of the diffusive line. The cathodes capacitance, distributed in half a cell to the right and half to the left of each tap, is supposed to be lumped at the tap itself, as shown in fig. 3. In this figure the considered noise generators are also drawn. See Appendix 3 for details. The other set, labelled as the "distributed model" refers to a more exact analysis in which the diffusive properties of the line are taken into account. The cathode and the connecting resistors are treated as a resistive line with a continuouslyg¢distributed capacitance. See Appendix 4 for details The terms A i in table 1 are written as tile product o f some factors:

2 2

A i - eNCD t~ Tp Ca G i F i '

a4rn .

a6,n: term due to the correlation between the

terms a 2 r n and

a4r n.

aTm : term due to the noise present at the two amplifiers outputs ruth and (m + 1)th as a consequence of the thermal noise of the line connected between them. asrn: term due to the correlation between the

for/=l

t o i = 6 , and

A i = kTCDGiF i ,

Even the diffusive line treated with the telegraphist equation is an approximate model because the excited waves are not TEM waves: however we have verified that the model is reliable, as far as the noise problems are concerned, if the "slimness" of the cell 2aiD is larger than l.

Table 1

22{

Tp

2

eNCD Tp

2

S) 2 P~

~[(m+l-S)

(m-

1

S) Pro_t] 2

S) Pm+I + ( m - I - S ) P m _ I ]

Pm+l+(m

~ ( m - S) 2 P2rn

(m

2 2 { eNCD NTa As=--c a f _ a ( m - S) Prn[(m+ l Tp

A4 -

-

_ eNCD

A 2 -

2 2 eNCD

A 1 - eNCD --~-p Ca2

_

Gi

} •

2 3

rD

-r F

6 ~ ---

oJ

~ U ( O , r F , rD) 20(20)1/2

sinai2 )

c°st2 )

-cosht2 ) cosh(20) 1/2 - cos(20) 1/2

2O 04 U(O"rF'rD) cosh(20) 1/2 - cos(20) 1/2

o f dO

do sinh(20) 1/2 - sin(20) 1/2 O-4 U(O'rF' rD) 20(20)1/2 c°sh(2O)l/2 c°s(20)1/2

r~3

- rF)2 0

1 2r F + 1

rr~ - r F ) 2

2

1

4

cosh(20) 1/2 + cos(2O) 1/2

-~U(O,rF, rD) 0 cosh(20)l/2 _ cos(20)1/2

dO

2 2rF+ 1

_2 __r D ? (1 - rF) 2 0

1 -- r F

?

rF)2 0

rD

-r F

~(1

1

1

2

~ U ( O , r F,r D) 0 2 .

~_D f do

~r(1-rF) 2 0

4

Fi

U(O'rF'rD) = 2 - 2c°s(I~rrDFO) +c°s(~D O) +c°s(~D) - 2 eOs[l t 2rD O)

sint2 )

Notes. (1) The terms Gi listed in the table correspond to the simple weighted centroid method. For the method based on the linear centroid finding filter, simply substitute the terms (m S) Pro, (m + 1 - S) Pro+1 and (m - 1 - S) Pro- 1, respectively, with the functions P(m S), P(m + 1 - S) and P(m - 1 - S). (2) The terms F i calculated by the distributed model are indicated by an asterisk. The terms F i calculated by the lumped model are indicated by two asterisks. (3) The function U(O, rF, rD) is:

2

o

2

( m - S) 2

1 - S ) PmPm+1

- S) 2 P2m+ I +

2(m-S)(m+

[(m + 1

A 8 = kTC D

{ P~m]

~(m-S)Pmi(m+l-S)Pm+l+(m-l-S)Pm-ll)

Gi

~

Tp

eNCD

2

1(continued)

A 7 = k TCD

A6-

Table

2] 2~F+1

[0~ 1/2

2O cosh(2O) 1/2 - cos(20) 1/2

[0~ 1/2

412r F +

(-b 1

(1 ~- r v ) 2 g 04 U(O'rF'

22rF+ 1 3 rD

cosh(20) 1/2 _ cos(20) 1/2

i o U ~ [o~~ lo? ~ [o ~ : c°sht2) sint2 ] + sinht~ ) c°st2 J

4 __r~ f dO sinh(2O) 1/2 + sin(20) 1/2 ~r(1 - rF) 2 0 ~ U(O'rF' rD)(20)l12 c°sh(20)l/2 - c°s(20)1/2

,t-7I

I(

F;

r~

332

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A'. Gatti et al. / Centroid m e a s u r e m e n t s in M W P C

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E. Gatti et al.

/

Centroid measurements in M W P C

for i = 7 and i = 8. In the above expressions the term eL is the physical power spectrum of the series noise generator of the amplifiers (under the assumption that all amplifiers are equal). The term ca is the FET input + feedback capacity normalized to the capacitance of the line between two contiguous taps (Ca = CA/CD). The constant h can assume the values 0, 1 or 2 depending on the considered A i (see table 1). The term G i (see table 1) depends on the weights [Pro or P(m - S)] and is the same in the lumped and distributed model. The factors Fi depend on the transfer functions of the system connected to the input of the amplifiers and on the current response of the amplifier-filter unit. It is given in closed form in the lumped model approximation, while only the integral form was possible in the distributed model theory. The meaning of the symbols used in table 1 is clarified at the top of the table itself and in Appendix 1. In figs. 4a to 4h the terms F i are plotted as functions of the shape- factor rv of the 6-current pulse response of the preamplifier-filter block (rv = T v / T p , see fig. ld). Each term F i is plotted for various values of the normalized time constant of the line (rD = R D C D / T p , where R D and C D are the resistance and the capacitance of a single cell of the line). In the figures the results are plotted of the theory based on the "distributed model" and those based on the "lumped model". For comparison, the terms corresponding to the case of isolated cathodes (see fig. l c) are also plotted. This case is the limit of the lumped model theory for R D -+ ~, that is for rD -+ .o (as far as the noise is concerned and neglecting the capacitance between contiguous cathodes). It is interesting to note that, as expected, the term Fa, due to the presence of the input capacitance of the amplifier, is coincident in the lumped and distributed theories. As expected, the term F2, due to the presence of the line impedance, is overestimated in the lumped model. In fact, this oversimplified model does not take into account the filtering action of the diffusive line on the high frequency component of the amplifier series noise. Obviously, the term F2 corresponding to the case of isolated cathodes, plotted in the same figure, does not suffer from this overestimation. The term F3 represents the correlation between Fl and Fz. The term F4, due to the noise charge injected into the contiguous taps, is only a little overestimated in

333

Table 2 V a l u e s o f G i f o r t h e w e i g h t e d c e n t r o i d m e t h o d . 3 t a p s active. P - t = P o = P1 = l , o t h e r P m = 0 G1 G2 G3 G4 Gs G6 G7 G8

= = = = = = = =

2 + 3S 2 2 + 3S 2 2 + 3S 2 2 + 8S 2 4S 2 4S 2 4 + 6S 2 2S 2

the lumped model. The terms Fs and F6 represent, respectively, the correlation between F1 and/74, and between F2 and

f4. The term F 7 , representing the thermal noise contribution of the line, is underestimated in the lumped model. The term F8 represents the correlation between the noise signals, due to the thermal noise of the line, at the contiguous taps. The terms G i in the simplest case in which P m = 1 for the considered amplifiers and P m = 0 for the others, have been evaluated by considering as active only three adjacent amplifiers. The results are listed I I I I I G7

G~= ~= 3

TAP rn

TAP m

Gs=G5

I TAP m+l

TAPm+I

Fig. 5. G r a p h s o f f a c t o r s G i a p p e a r i n g in t a b l e 1 versus t h e a v a l a n c h e p o s i t i o n S: (a) t h r e e t a p s m e t h o d ; (b) c e n t r o i d e v a l u a t e d b y t h e s y s t e m o f fig. 2a.

334

E. Gatti et al. / Centroid measurements in MWPC

in table 2. We note that the terms Gx, G2 and G3 are identical; the same holds for the terms Gs and G6. The t e r m s G i are plotted in fig. 5a, In fig. 5b the terms G i are plotted corresponding to the more elaborate weighting technique based on the linear centroid finding filter.

zoF.-

0

O.~

0.6

OB

IO

L5

For evaluating e~ according to eqs. (2) and (4) we need to calculate the terms

or

.~:F=O B

Vrn

~F=I] t,

1

(8)

respectively. The signal Vm at the output of the ruth preamplitier-filter unit can be written as

Vm = qREwooMT(rn - S) .

OZ

IF

4. Evaluation of the signal terms in the variance e~

d ( ~m Vm - ~ P( rn - s)) 2 ,

_J

(9)

In this equation, q is the avalanche size. The term REw~ is given by eq. (8) and fig. 3 of ref. 1. REw~o , represents the charge induced in all cathodes at t = 6.389To [see eq. (4) of ref. 1] normalized to the avalanche charge. The term M is the peak of the output pulse of the preamplifier-filter unit when the input current pulse is 1/2(t + To). This pulse is the current delivered by a proportional counter normalized to unity induced charge (at t = 6.389To) and to the characteristic time To. In fig. 6a the factor M is plotted as a function of the shape-factor ~.. of the trapezoidal response and for different values of the ratio Tp/To. The waveform urn(t) at the Output of the amplifier-filter unit is plotted in fig. 6b for different values of the parameter of the trapezoidal response . Note that in the calculation of these responses the effects due to the propagation time in the diffusive line are neglected. In this paper we use the same M as in ref. 1; even if the filter considered in this paper has a maximum amplitude of 1 while the corresponding filter of ref, 1 differs because the peak of its integral has the value of ½. We compensate this fact by writing in this paper the maximum output amplitude due to an avalanche q as q R E w o ~ while in ref. 1 the maximum amplitude is ½qREu,~ll. These differences are however immaterial and are compensated in the results which are ratios of signal to noise squared.

z l:ig. 6. (a) Graph of t:actor M proportional to output pulse peak amplitude in eq. (9) versus shape factor rl,,. Tp/T o is the ratio of the f-current pulse response width to the characteristic time T O of the proportional counter. (b) Output pulse shapes, r is time normalized to ,S-current pulse response width Tp. r F is the shape factor of the 6-current pulse response.

The term 7(m S) is a weigl~ting function depending on the distance m - S between the avalanche position and the considered tap mth. This function can be easily evaluated from eq. (42) in ref. 1 taking into account that the charge induced on an infinitesimal element of the diffusive line is shared linearly between the contiguous taps, according to the well known charge division rules. The weighting function 7(m - S) has been calculated for the taps - 1 , 0 and +1, taking the tap number zero as the origin of the coordinates, and is plotted in fig. 7a for s/D = 0.5, D/R = 666, 2aiD = 2.5. The function 7(m - S) indicates the fraction of the induced charge that is collected at the ruth tap. In the simplest case in which all Pm = 1 in the range in which the induced charge is not negligible (not more than three cells), the term (7) reduces to:

= qREwJl/I

7(m

S)

= (qRt:.'w~M) 2 . (10)

For the technique based on the linear centroid finding filter, the term (8) has been evaluated by calculating the derivative of the weighting function P(m - S), plotted in fig. 2c, and by calculating the values Vm with eq. (9). The term (8) can be written as

335

E. Gatti et al. / CentroM measurements in MWPC 10

By using the notation of the preceding sections we can write

08

( q R E worM) 2 r~2

(G 1F1 C2a + G2 F2 Ca

01,

+ G3F3 + G4F4 + G s F s + G6F6)

02

+ kTCD(GTF 7 + G s F s ) )

\ $

O0 ~z

O

02

O3

01,

S

l

=

0.5

2

2

( CN CD

(qREw~M)2rl2\ rp

o~+ kTCDi3} .

(13)

108

c~ and/3 can be evaluated, by using the graphs of the preceding sections, for the requested values of the parameters rv, rD, Ca, To and for every avalanche position. As an example, the terms c~ and/3 are plotted in figs. 8a and 8b, respectively, for the simple case of weights P-~ = Po = PI = 1 and other Pn = 0. For the linear centroid method they are plotted in figs. 9a

I Or.

1 O0 096 0.92 0 81

o'1

o'.z

o'.3 o14 S os

Fig. 7. (a) Graph o f the function ~, which shows the sharing of the induced charge among three contiguous amplifier inputs versus avalanche position S. (b) Graph o f the factor ~2 of eq. (12) versus avalanche position S.

OISTRIBUTEDMODEL/ / %=1Z~ ,SOLATEDCATHODES ~ , . ~ / _ ~ % / / ~,~

d/'(m - S).) 2 ×

dS

"

(11) THREE TAPSCENTROID

The terms (7) and (8) can thus be rewritten as

I

(qREw~oM) 2 772 ,

02

(12)

I

O.t,

I

0.6

0.8 "cF

where 7/2 = 1 for the case of term (7), and 1Z DISTRIBUTEDMODEL 10

ca-1

"%:1 I

for the case of term (8).The term rl 2 of the latter case is plotted in fig. 7b for s/D = 0 . 5 , D / R = 666, 2aiD = 2.5. As can be seen it is nearly equal to 1 for every S. This fact also holds for other geometries (2a/D) and the congruent width of the interpolating filter. THREE TAPS CERTROIO

5. Evaluation of the variance e} For an easy evaluation of the variance e~ in various practical situations we will give some other useful graphs in this section.

0'.2

0!~

°!6 "c~ o.8

Fig. 8. (a) Graph of a in eq. (13). ~ is proportional to the centroid variance contribution due to amplifier series noise. (b) Graph of j3 in eq. (I 3). /3 is proportional to the centroid variance contribution due to thermal noise. Centroid evaluation with three taps active.

336

E.

1I

G a t t i et al. / C e n t r o i d m e a s u r e m e n t s

Table 3 Parameters of the MWPC. Refer to fig. 1 for the meaning of the symbols used

///

z 0=

L = "25 cm D = 1 cm 2 a / D = 2.5 (first example); = 1.2 (second example)

/// 30 20 10

0~

b = s/D = 0.5 D / R = 666 (R

55 ,i 02

0/,

O.fi

J

q = 106 electrons (size of the avalancbe) e~ = 3 X 10 -18 V 2/Hz (amplifier series noise)

"cF O.B

~I

2

LINEAR CENIROIDFILLER O

i

02

i

O~

I OB

= anode radius) = 1.2

r D = RDCD/Tp ca = C A / C D = 1

DISIRIBUIED MODEL

10 ;0=1.z c.=I

in M W P C

%F

0.8

Fig. 9. Centroid evaluated by system shown in fig. 2a. (a)

Graph ofc~ in eq. (13). (b) Graph ofp in eq. (13).

and 9b, respectively. In both cases a and 13are plotted as functions of ~-v and for some values o f the avalanche position. The typical values of ~-D = 1.2 and Ca = 1 are considered in this example. The value o f a for the case of isolated cathodes is also indicated in fig. 8a.

6. Conclusion and remarks. A comparison between the multi-tap resistive division method and the isolated cathodes method We consider an MWPC, whose parameters of interest are summarized in fig. 1 and table 3. As a first example, we consider the case in which the ratio of the intertap-distance 2a to the anode p l a n e - c a t h o d e plane distance D is equal to 2.5. The capacitance of a single cell of the diffusive line is about 20 pF. It is interesting to evaluate the variance e} as a function of the resolving time Tp. By using eq. (13) and tile appropriate graphs of the preceding

sections we evaluated the variance e~ in the simple case of three active taps at a time with weights equal to one for S = 0. The results are plotted in figs. 1 0 a - c for the characteristic time To equal to 2.5 ns, 25 ns and 100 ns, respectively. In these figures, the value of e~ is plotted in the case o f resistive line connecting the cathodes and in the case of isolated cathodes. For the first case the values of e~ are also plotted due to the amplifier noise only and due to the thermal noise o f the line only. It can be observed that tile value of e~ for the isolated cathodes is always higher than the corresponding value for the diffusive line due to the amplifier noise only, as expected, because of the capacitive load lumped at the amplifier taps. With long resolving times, the variance due to the thermal noise is higher than that due to the amplifier noise. It can then be seen from figs. 1 0 a - c that, as expected, with short resolving times the diffusive line method is advantageous, since it results in a smaller variance. Conversely, at high resolution times the isolated-cathodes method is better. The cross-over point is a function of the characteristic time To, assuming constant values for all the other parameters. The quite surprizing fact that a method which adds thermal noise through the resistors of the diffusive fine can offer better resolution than the method without resistors, is explained by the opposite effect o f tile presence o f the line at the input of the amplifiers. In fact, the series noise generators o f the amplifiers lose effectiveness because they are connected, at high frequencies, to the higher impedance of the diffusive lines instead of to the lower impedance of the lumped capacitance of the isolated cathodes of width comparable with the cell dimensions. in fig. 10d the variance e} is plotted, for a characteristic time To = 2.5 ns and for the parameters of

337

E. Gatti et aL / Centroid measurements in MWPC 100

~ ~

To = 15

ns

a

\ \- -

10 "1

10-1

10-i

lO-Z ......

10-

10~

10o

DIFFUS VE LINE

~"%..',..

10-3

,SOLATE0 :AT,°0E°

"

T,HEE TAP° ENT,,,0

TO= Z5ns

103

10-~ 10

~,,~

~.~~~ /50s

90ns

TO:lOOns

C

10-

lO-t

'2"

THREE TAPS CEINTROID

10"

b

10z

10-3

~X \

10~

Tp(ns)

10~

10"~1THREE1'0z TAPS EENTROID103

Te(ns)

1;4

Tp(ns)

10°

100 X

To=2 5ns

d

\

TO= ?5 ns

e

10-1

10-I

~k,-oZ8ns

\ lO-Z

10-t

10-3

lO-J

~.

°.

\ -,),,,

\\

LINEAR CENTROIOFILTER 110

THREETA,S :EHT.,0

llo;r

103 Te(ns)

10-~

103

Tp(ns)

Fig. 10(a)-10(c). Total relative centroid variance e} for three different values of the characteristic time To of the proportional counter. "Cell" geometry a i d = 1.25. Three taps method. (d) Total relative centroid variance e} for a single value 2.5 of the characteristic time of the proportional counter. "Cell geometry aid = 1.25. Centroid evaluated by the system of fig. 2a. (e) As fig. 10a but for a different geometry of the "cell"; aid = 0.6. table 3, in the case of the linear centroid finding

filter. We note that the resolution is not very different from the corresponding case of fixed weights (fig. 10a). The advantage of the method is mainly the high differential linearity. It is worth noting that the geometry of the MWPC here considered (aiD = 1.25) is very far from the optimum conditions, in terms of resolution and linearity, for the case o f isolated cathodes. For an optimum case o f isolated cathodes the results are given in ref. 1 in figs. 9a, 9b, and 1 2 a - c (for three taps active). A reasonable linearity could be obtained by the interwoven cathode method [2].

As a second example, for a more significant comparison between the isolated cathodes and the diffusive line methods it is convenient to consider a geometry o f the MWPC which gives acceptable results in terms o f linearity and resolution for the conventional isolated cathodes. According to ref. 1 we can impose a ratio a i d = 0.6. In this case, the capacitance CD of the cell is reduced by about a factor o f two. By considering eq. (13), we can see that, if the ratio c a = C A / C o remains equal to one, the term depending on the amplifier noise is reduced by a factor four while the term dependent on the line noise is reduced by a factor two. In fig. 10e the variance e~ is

338

E. Gatti et al. / Centroid measurements in MWPC

plotted for this geometry, for To = 2.5 ns and for the case of constant weights. We remember that the variance e} is given assuming the intertap distance 2a as the unit o f the length. In order to obtain the effective resolution we must thus multiply the calculated e s by the distance 2a. In the present work the considerations about the optimization of the geometry of the MWPC in terms o f the ratio "cathode w i d t h - a n o d e plane to cathode plane distance" [1] do not appear. In fact, in the present work the calculated variance is the variance of the estimator, while in ref. 1 we considered the variance of the avalanche position. The values of the two variances only coincide with geometries which have a good differential linearity, as, for example, for isolated electrodes with an interwoven system having an ideal triangular distribution of induced charge or for the diffusive method when all the charge is induced on two adjacent cells (for the chosen overall width of the centroid finding filter of approximately five cells).

Appendix 1 - Table of symbols a -- half width of the elementary cell of the MWPC (see fig. 1). aim - a8m = terms, except for a proportional factor, in which the noise contribution o f a single noise source can be split (that is the series noise E m of the ruth amplifier, or the thermal noise o f the resistive line between the taps m and m + 1). A l - A8 = terms in which the "noise term" of the variance can be split (that is the numerator of the eq. (2) or eq. (4)]. b m ( t ) = inverse Laplace transform of Bin (p). BIn(P) = term proportional to the transfer function that relates the fluctuation dSm o f the estimator S o f the avalanche position to the noise signal Era. c = capacitance per unit length o f the diffusive line connecting the taps of the readout system. Ca = CA~Co = input capacitance CA of the amplifier, normalized to the capacitance CD of an elementary cell o f the diffusive line. C A = C i + Cf = input capacitance of the amplifier. It is the sum of the FET input plus the feedback capacitance. CD = capacitance of an elementary cell of the diffusive line. Cf = feedback capacitance of the amplifier.

C i = parasitic capacitance of the input FET o f the amplifier. dS = fluctuation of the estimator S of the avalanche position, due to all the noise sources. d S m = fluctuation of the estimator S of the avalanche position, due to a single noise source (that is the series noise E m of the mth amplifier, or the thermal noise of the resistive line between the taps m and m+l). D = cathode-plane to anode-plane distance in the MWPC. E m = series noise voltage o f the ruth amplifier. Ew= = term proportional to the charge induced in all cathodes at t = 6.389To, normalized to the avalanche charge [see eq. (4) of ref. 1 ]. fL(X) = ratio of the current injected in the amplifier at the left of the "elementary cell", when a current generator Io is connected at the coordinate x of the line (as shown in fig. 1 lb). fR(x) = as for fL(X), but with reference to the right amplifier. Fi = one of the factors in which the terms A i c a n be decomposed. The factors Fi are listed in table 1. G i = one of the factors in which the t e r m s A i can be decomposed. The factors G i a r e listed in tables 1 and 2. h = integer, that can assume the values 0, 1 or 2, that appears as exponential o f Ca in the expression o f A i. h(t tin) = ideal &response of the linear centroid finding filter. I m = equivalent current-noise generator of the ruth amplifier. Ira,m+ l = Thermal noise generator of the resistive line between the taps m and m + 1, respectively. l ( x ) = current flowing in the diffusive line at the coordinate x. Io = current generator used in order to calculate the functions fL(x) and .fa(x). IL(X) ; current flowing in the virtual ground o f the left hand amplifier of the "elementary cell" when a current generator 1o is connected between the coordinate x and x + dx o f the cell, as shown in fig. l l b . IR(X) -- as for IL(X), but referring to the right hand amplifier. K = Boltzmann constant. L = side dimension of the MWPC. m = integer, means the ruth amplifier, and its coordinate (when the distance between two contiguous taps is taken as the unit length).

E. Gatti et al. / Centroid measurements in MWPC M = factor proportional to the peak o f the output pulse o f the amplifier-filter unit. p = complex frequency in the Laplace transform domain. Pm= weight that is attributed to the signal at the tap m for the centroid evaluation in the "finite number of active taps method". The value o f Pm is a constant, for a given range of the avalanche position.. P(m - S) = weight as Pro, in the "linear centroid filter m e t h o d " . P(m - S) is a function o f the distance between the mth considered tap and the avalanche position. q = avalanche size. r = resistance per unit length o f the diffusive line connecting the taps of the readout system. R = anode wire radius. R D = resistance o f an elementary cell of the diffusive line. s = anode wire spacing in the MWPC. S -- estimator of the avalanche position. t = time. t m = time o f the sampling of the m t h tap signal, with the linear centroid filter method. tf = zero-crossing time o f the output of the linear centroid filter. Ti., TR, Tp = parameters o f the 6-current response of the amplifier-filter unit (see fig. 1 d). To = characteristic time of the proportional counters in the MWPC [see eq. (4) of ref. 1]. T = absolute temperature. T s = period o f the sequential switch-and-hold in the linear centroid filter method. u(t) = step function. U(O, zv, rD) = function proportional to the Fourier transform of the (5-current response o f the amplifier-filter unit. It appears in table 1. vm(t) = signal at the output of the rnth preamplitier-filter unit. Pin(t) = signal at the input of the linear centroid finding filter. Vm = peak amplitude of the signal at the output of the ruth preamplifier-filter unit. zl, z2, za = terms that appear in the computation o f the variance of the estimator S, with the distributed model. Z = line impedance seen from the input of the charge-preamplifier. w(t) = 6-current time-response of the preamplif i e r - f i l t e r unit (see fig. 1 d). W(jw) = Fourier transform o f w(t). x = space-coordinate along the diffusive line. x =

339

0 at the mth tap, x = 2a (= 1 if the length of the elementary cell is normalized) at the tap (m + 1)th. a = term appearing in the expression of the variance e~ [see eq. (13)]. The term a is proportional to the variance contribution due to the amplifier series noise. /3 = term appearing in the expression of the variance [see eq. (13)]. The term t3 is proportional to the variance contribution due to the thermal noise of the diffusive line. 7 ( m - S) = sharing factor of the induced charge among the amplifier inputs. 6 = impulse function. e~ = variance o f the estimator S of the avalanche position, due to all the noise sources o f the readout system (diffusive lines and amplifier noise). e 2 = variance of the estimator S of the avalSm anche position, due to a single noise source (that is the series noise Ern o f the mth amplifier, or the thermal noise o f the section o f resistive line between the taps m and rn + 1). e~v = physical power spectrum of the series noise generator o f the amplifiers (under the assumption that amplifiers are equal). 7? = factor appearing in the denominator (signal term) of the expression o f the variance e~. 0 = COTD. Angular frequency normalized to 1/TD. j = imaginary unit. p = width of the function h(t), normalized to the period T s . rF = TF/Tp parameter of the 6-current response of the amplifier-filter unit (see fig. l d), normalized to the time Tp. rD = R DCD/Tp time-constant o f the diffusive line, normalized to the time Tp. ~ x ) = Laplace transform Of the potential at the coordinate x of the diffusive line. co = angular frequency. Appendix 2 - Estimator S of the avalanche position and its variance

(a) The estimator S of the avalanche-position can be assumed to be a weighted-centroid of the peak amplitudes Vm of the signals at the output of the amplifier filter block: J

S : ~ m t ' m V m / ~ P m Vm .

(14)

The variance e~ of the estimator S can be evaluated as follows. As the estimator S is a function o f the signals Vm, we have that

E. Gatti et al. / Centroid measurements in MWPC

340 aS

dS = G - ~ m

(15)

dVm.

From tile eq. (14)

_ (17l

S)Pm

G Pm Vm consequently:

dS = ~ (m - S) Pm d V m / ~

PmV,,, .

(16)

The variance e} is the average value of dS 2 6~ = dS 2 . From eq. (16) we get

- S)(k-

S)PmPk

X clVm dVk .

(17)

(b) Linear centroid finding filter. Refer to the fig. 2 o f the main text. We first suppose, for the sake of simplicity, that the signal at the input of centroid filter is:

GVm~)(t-

tin),

(18)

t m = niT S , where Ts is the time interval between the sampling of two contiguous taps. Let the 8(t - tin) response of the filter be the function h(t tin):

h(t - tm) = ( @ X [u(t -

t.,)

tr

~mV.,

.

Ts

~lZ m

2

(21)

The normalized time of zero-crossing is thus given by the centroid of the signals Vm plus a constant term #/2. We assume .S = tf/T s - ~/2 as the estimator of the avalanche position. The output of the filter is the sum of the signals Vm linearly weighted with a weighting function P(m - S). This filter output has a zero crossing corresponding to the centroid of the signals Vm, with a constant time delay. P(m - S) is the space-domain replica of h(t tm), with a normalized intertap distance. The space-domain dual of eq. (20), that is

VmP(m -- S) = 0

e} - ( ~ p , l n v , , , ) 2 ~ ( m

Oin(t ) =

We get

u ( t - t m -- u T s ) ]

,

(191

where /2 is the width of the response normalized to the time T s. The slope of this function is equal to one for tm < t < t m + liT S. The zero-crossing time tf of the output of the filter is given by

Vmh(t f - t,n ) = 0 , that is by

G Vm(e-~ (tr- rots)) = 0.

can be used as implicit definition of the centroid S. A more realistic 6-response of the filter is drawn in fig. 2b of the main text and is used in the present work. When the input signal of the centroid filter is

vin(t)

= ~

VmlU(t tin)--u ( t

(20)

--

tm TS)I

(23)

as in the case of fig. 2a, the resulting weighting function is, obviously, the convolution of the 8-response with the rectangle function. This function has been digitally computed and drawn in fig. 2c. From eq. (22), we can obtain:

~(V,n

+dV,,,)P(m -- S

dS)=0.

(24)

And then

(VmP(m

(t - m T s ) )

(22)

S) + V m

+ dVmP(m - S )

OP(m - S)

+ dV m

aS aP(m

dS

~ S ~

3S

ckS'I = 0 .

The first term of the sum is zero, the last one is second order term. We have:

dS = ~ dVmP(m - S ) / ~ i

V,,,

3P(m

a

S)

aS

From which:

e2s=dS 2= ~P(m S) P(k- S)dVmdV k (~Vm Ot'(m_as-S).)2

(2s)

E. Gatti et al. / Centroid measurements in MWPC

Evaluation of the variance eSrn' 2 • From eq. (16), that we recall here as

Appendix 3 - lumped model (a) Evaluation of the contribution of the series input no&e of the amplifiers to the "noise term" [eqs. (5) and(6)] We refer to fig. 3 for the diagram of the lumped model and for the symbols used. According to the main text, we first suppose that only the noise generator Em of the mth amplifier is active. In this case only the signals Vm_ 1, Vm and Vm+a are affected by fluctuation dVm-1, dVm and dVm+l due to the input noise. Evaluation of d Vm : the line impedance seen from the input of the ruth amplifier is: Z = ~D

341

dS ( ~ V i P i ) =

~(i-S)PidVi

and from eqs. (29) and (30), we have:

+ (m - S ) P m p C A WE m

+ [(m - 1 - S ) Pm-1 + (m + 1 -S)Pm+l]

× - ~

WEm .

(31)

From the preceding considerations the meaning of the three terms in the above equation is obvious. If we write eq. (31) with a synthetic notation

+ PCD

As W(jcJ) is the current transfer function of the preamplifier-f'dter block [see in fig. l d the 6-current time response w(t)], the fluctuation dVm is easily determined by considering the Laplace transform I m of the equivalent current-noise generator of the mth amplifier

dSm ( ~ ViPi) = Bm(p) Em

and if we suppose that the noise generator E m has a white spectrum, whose physical power density is e~, we can write 1

I m =(I+PCA)E m

(27)

where:

(32)

Sm

ViPi

e~

--f- f -

+~

[bm(t)12 dt,

(33)

where bm(t ) is the inverse Laplace transform of B m (p). From eq. (31) we have

C A =C i +Cf.

dw(t)

The fluctuation d Vm is simply given by: 1

dV m = 14/1m = -~ WE m + pC A WE,n .

bin(t) = (m - S) PmCA - dt (28)

+ ( m - S ) Pm(~D w(t)+CDdw(t)]dt ! In the above expression, the term 1

+ [ ( m - 1 - S ) Pm_ 1 +(m+ 1 - S ) P m + l l

WEre

takes into account that part of the effects of the generator E m upon the output mthdue to the presence of the line. The term

We can calculate:

p C A WEm

Ibm(t)[2 = (m - S) z PmC#, 2 2 Iw'(t)l 2

takes into account the part due to the presence of the input capacitance of the amplifier. From eqs. (26) and (28) we get

d Vm =

+ PCD WEre +PCA WEre •

(29)

Evaluation of dVm_l and d Vm+l: the fluctuations are simply 1 dVm_ 1 = dVm+ 1 -

RD

WE m .

(30)

+(m- S)2P~ml~---~w(t) +CDW'(t)l2 2 + 2(m - S)2P2nCA(-~D w(t) w'(t) + CDIW'(t)I , 2) + [ ( m - 1 - S ) Prn-1 + (m +1 - S ) P m + l ] 2 1 × R--~) lw(t)[2

342

E. Gatti et al. / Centroid measurements in MWPC

+ 2(m - S ) PmCA [(m - 1 - S ) Pro-1

+(m+ 1 --S)em+l]

/ + ~ Iwl z d t = T P + 2 T ~ ' f+ ~ Iw'[ 2 d t - -4 _oo 3 ' ,_~ Tp- Tv

~ W(t)w'(t) 6')

(37)

+ 2(m - S ) Pm [(m - 1 - S ) Pro-1 + (m + 1 - S ) Pm+I]

1

2

w'(t)).

By recalling that, if w(t) = 0 for t -+ + ~ +~

/

w ( t ) w ' ( t ) dt = 0

(35)

_co

(which corresponds to the well known fact that a function w(t) and its derivative w'(t) are uncorrelated), from eqs. (33) and (34) we obtain:

By substituting eq. (37) in eq. (36) and by a suitable change of symbols as listed in table 1 of the main text, we can obtain the terms air n ,..., a6rn as expressed in table 1. Now, the variance e 2 just obtained is due only to Srn the input noise generator of the ruth amplifier. As the noise generators of the amplifiers are uncorrelated, in order to obtain the variance due to all the amplifiers we must simply sum the variances due to each noise generator. In this way we can obtain the terms A1 A6 of table 1. Note that if, instead of using the simple weighted centroid method, we use the linear centroid method, the only change that we have to make in eq. (36) is to substitute the terms (m - S)Pm with the functions

t'(m - s). m -- S) 2 PmCA 2 2

=

Iw'l 2 dt

2

2 /4 + (m - S) Pm('-~D f

Iw[ a t

(1)

+~

+C~) f lw'ldt)

(b) Evaluation o f the contribution o f the thermal noise o f the line to the "noise term" [eqs. (5} and (6)] According to the main text, we first consider the noise generated by only the cell of the line between the taps m an m + 1. That is, we only consider the generator I m,rn+l of fig. 3. The consequent fluctuations at the output are:

(II) d Vm = - W l m , m + l ,

_oo

dVm+ 1 = +Wlm,m+ 1 •

+~

+ 2(m - S ) 2 p 2 C A C D f

(38)

Iw'[2 dt

(II1)

From which :

_oo

+ [(m - 1 - S ) P m - 1

x

1

dSm ( ~

+ (m + 1 - S ) P m + I ] z

+ (m + 1 - S ) P m + 1 W l m , m+l .

+~

f Iwt: dt

ov)

+0 +(m-S)Pm[(m-

ViPi) = - ( m - S ) Pm WIm,m+l (39)

Let 4 k T / R D be the physical power density of the thermal generator. We have

(v) e2 Sm

1 - S ) Prn-1

ViP i

=

R I)

+~

+(m + , ,),m+,l (

L

d,1

{V,)

X .(

[ - ( m - S ) P , n w + (rn + 1 - S ) Pm+lW[ 2 dt

(36) The terms I to VI in the above expression are called airn to a6m , respectively in the main text. We refer to the main text for the physical meaning of the six terms. We recall that, in our case (see fig. 1 d),

_ 2kT ( RD [(m + 1 -S)2p2n+ 1 + (m - S)2P~z] +oo

X /

[wl2 dt

E. Gatti et al. / Centroid measurements in MWPC

f Iwl2dt)

-2(m+l-S)(m-S)Pm+lPrn

_oo

The first and the second term in the above expression are called a7m and a8m , respectively, in the main text (refer to the main text for their physical meaning). In order to obtain e~ due to the complete line we must sum the e Srn 2 due to the single cells of the line. The noise generators of the single cells are uncorrelated. In this way we obtain the terms A 7 a n d A s of table 1.

(a) Evaluation o f the contribution o f the series input noise o f the amplifiers to the noise term [eqs. (5) and (6) As done with the lumped model, we first consider active only the noise generator E m . Evaluation of dVm: we must determine the impedance of the distributed line as seen from the amplifier input. Let 2a = 1 be the length of the line between the mth and (m + 1)th taps, assuming the ruth tap as the origin of the coordinates. The element dx of the line may be represented as in fig. l l c . Let ¢(x) be the Laplace transform of potential at the coordinate x, and I(x) the transform o f the current in the resistive line at the same coordinate x. By imposing the boundary conditions:

(a)

-Io~l,

-

i

!+lol,

x

.

x+Ax

(b)

~ x

(41)

we obtain, after well known computations (r]

4)(o) = ~pc l

1/2 I(2a) sinh[(pre) i~z 2a] , (42)

I(0) = I(2a) cosh [(prc) 1/2 2a] . The impedance of the line between the taps m and m + 1, seen from the tap m, is easily determined from eq. (42) .

1 (RDCD~ '/2 Z = 2CD \ - - 7 - - - / tanh(pRDCD) 1/2 .

(c)

dx Fig. II. (a) and (c) diffusive line as limit of lumped elements; (b) current "doublet" injected in the diffusive line.

(44)

Now, following the procedure adopted in the case of lumped model, we obtain 1

d Vm = -~ WEre + p C A WErn 2CD = 'TD (.pTD) 1/2 coth07TD) 1/2 WE m +PCAWEm (45) where To = R D C D . For the physical meaning of the two terms see the Appendix 3. Evaluation of dVm_ 1 and dVm+l: From eq. (42) it follows that (46)

As 4>(0) = -Era, we find dVm+l = dVm-1 -

CD WEm(pTD)I/2 TD sinh(pTD) 1/2 .

(47)

Evaluation of the variance e~: from eq. (16), we get:

I(x)+dl(x) Icdx

(43)

Taking into account also the line between the tap m an m - 1, the total impedance seen from the tap m is:

1

(x) d~(x) J-f-- - - ~-- ~(x)+d~(x) l(x)

=_ri(2a), 2a

CD 1 l(2a) = ~ (pTD) '/2 sinh(.PTD)a/z ¢ ( 0 ) .

~_I Io

~

:

\ox/

~(°) =(~) '/2 tanh [(prc) x/2 2a] I(0)

Appendix 4 - distributed model

isl~ -

~(2a) -- 0 ,

d(, } (4o)

343

+ (m - S ) P m

2Co (pTo)l/2 cosh(pTi)) 1/2 TD

+ [(m - 1 - s ) e m - 1

sinh(p T D) 1/2

+ (m + 1 - S ) Pm+I]

344

E. Gatti et al. / Centroid measurements in MWPC

X \ - ~D-D] ~ i / 2 [

(48)

WEre

with the now well known physical meaning of the three terms. In order to evaluate the variance e2Sm due to the input noise Em it is now more convenient to perform all the calculations in the frequency domain. With the same symbols used in Appendix 3, we recall that (

e2Srn G V i P i

)2

e~

=T /

+~'

dco.

IB(CO)I227r

[z,I 2 = ( m - S )

2 P ~ cI T~

02

cosh(20) 1/2 "l- c0s(20)1/2 cos(20) 1/2

4CB

I~1~ = ( " - s ) 2 G r T o cosh(20) 1/2

(11) . 2 Re[zlz2] = ( m -

S)2PZm

0

+ (m + 1 - - S ) Pro+l]

2 Re [z, z3] = - ( m - S) tOm[(m

2C D cosh(j coTD) 1/2 + (m - S) Prn ~D (JcoTD)I/'2sinh(jcoTD) 1/2

--

1 --

{IV) S ) Prn-1

2CACD 0(2011/2 ~ ,

+ (m + 1 - S ) P m + I ] 1 + (m + 1

{III)

2C~ 1 X T-T-D 0 cosh(20)l/2 _ cos(20)l/2

(m - S ) tomjcoC A

+ [(m - 1 - S ) P m

Y~)

X 0(20) 1/2 sinh(20)l/2 - sin(20)l/2 cosh(20)l/2 _ cos(20)1/2 [z312 =[(m - 1 - S ) P m - 1

= e~

2CACD

(49)

We have. e2Sm (kG giP i

(i)

-S)Pm+I]

/ o u ,~

sinht-~)

X(-~-P-D)(JcoTD)l/2sinh(jlrD)l/2] 2

i o l , ,2

cost~ )

/ovJ2

- cosh[2)

× cosh(20)1/2 _ cos(20)1/2

X IW(jco)l 2 dco (50) 27r Let us write the preceding equation, with a synthethic notation and with a change of integration variable 0 = c o t D as

Sm

NiP i

= e~

2 Re[z2G]

= -(m - S)Pm

+(m+l-S)Pm+ll

lO~1 j2

sint}]

(v)

[(m - 1 - S ) P m - i

4cg T~

IZl(jO) + z2(jO) 0

dO + z3(jO)121W(jO)l 2 - -

2rrTD

(51)

,

where zt(jO) = (m - S ) P m CA jO , TD z2(jO) = (m - ' S ) Pm

2CD c°sh(j0)l/2 TD sinh(j0) 1/2

z3(jO) = [(m - 1 - S ) P m - l

00)1/2

× 0 cosh(20)t/2 _ cos(20)x/2 •

(VI)

(52) Note that the terms from I to VI correspond to the terms aim . . . . , a6m of table 1. For the case o f linear centroid method substitute the function P(m - S) for the term (m - S)Pm. We must now calculate the Fourier transform of the 8-current response w(t). We have then, with 0 = coTD

+ (m + 1 - S ) P m + I ]

(

1

X - ~

sinh(j0)l/2 "

By recalling that: Iz~ +z2 +zal 2 = Iz~l2 + Iz2l2 + Iz3I2 + 2 R e [ z l z ; ] + 2 Re[zlz3] + 2 Re[z2z3] and after some tedious calculations with the hyperbolic functions we obtain

[W(j0)[ 2 =

+cos(TIc~o

2 - 2 cos

2Tt

+cos(TP 01

\TD]

\TD

- 2 cos( T R + T F \ TD

0)1.

]

(53)

Now, with suitable change of variables (see the list of symbols in table 1 of the main text) the vari-

345

E. Gatti et al. / Centroid measurements in MWPC

ance can be expressed as a sum of six integrals of functions of 0 e Sm 2 =e~ f o

+e f o

]Zl[ 21W[2

dO 27rTD

where TD = R D C D is the time constant of the line. The derivatives o f f L ( x ) and fp,(x) are: 3fL(X)_ _ (PTD)I/2 cosh[(pTD)l/z(1 - x ) ] 3x sinh(PTD) 1/2 ' (}fR(X) - (.pTD) 1/2

3x

iz21=lWi2 __d0 + ..... 2aTD

= alto + a2m + . . . . . .

(54)

The six integrals are listed in table 1. Also in this case, in order to obtain e} due to all the uncorrelated generators Ern we must simply add up the e 2 Sm " (b) Outline of the calculation of the thermal noise The diffusive line cell between two contiguous taps can be represented by the limit for an infinite number of subcells of the lumped model shown in fig. 1 la. The current generators, shown in the same figure, represents the thermal noise of each resistor element rdX. A single noise generator and the consequent currents injected in the left and right amplifiers are shown in fig. 1 lb. Let us call, fR(X) =IR(X)/Io

oo=~ do0 f ~ [W(jco)[ 2 [(m - 1 - S) Pm-1 oJL Ox co=0 + (m - S )

~)fL(x) dx

IL =1o ~

3fR(x)

4kT dx r

(59)

/

~

IW(jco)l 2 P(m - 1 --s)3fLDx

¢O=0

+P(m

S) 3fR 2 4kT dx

(60)

-7for the case of the linear centroid finding filter. As the contribution of dx is uncorrelated to all the other contributions, we can integrate over all dx from 0 to 2a = 1 and sum the contribution of all other cells. We obtain

4D =G f

vo== dw

c~=O

,

2

for the simple case of weights Pm equal to one or zero, and as

(56) ×

I R =1o ~ X

Pm 3fR

(55)

the relevant transfer functions. We shall calculate I L and I R due to a current - I o and x and Io at x + dx by the so-called "transfer function field method" [5]. In the present one-dimensional case, the method reduces to evaluating the currents injected at the two contiguous virtual grounds, by means of the first derivative OffL(X ) and fR(x):

(58)

sinh(pTo)l/2

The contribution to the variance of the avalanche position estimator due to the thermal noise of the element rdx can be written as

in the diffusive line model

fL(X) =IL(X)/Io ,

cosh [(PTD) 1/2 X]

4kT

fwo )l _

RD

(2COTD)i/z

cosh(2cOTD)I/z _ COS(2OOTD)I/2

dx .

In the following, the spectral physical current density 4kT/rdx will be attributed to the current generator associated with the element dx of the line. The explicit form o f f L ( x ) and fR(X) for the diffusive line is

x{A22B----~2Isinh(2COTD)'/2 + sin(2WTD)'/2 F -

[ W T D ~1/2

2AB Lcosh t ~ - - ) [C.,.)TD~1/2

f L ( X ) = sinh [(pTD)I/2(1 -- X)]

+ sinh ( - ~ - )

sinh(PTD)l/2 sinh [(pTD) 1/2 x] f R ( X ) - sinh(pTD)l/2 ,

(57)

[COTD] 1/2

sin[ - - ~ - ] [o.)TD~I/2-]-])

c°s[`T)

JJ/'

(61)

where, for the simple case of weights Pn equal to one or zero:

E. Gatti et al. / Centroid measurements in MWPC

346

A=(m-

1-S)Pm_l,

D=~

B=(m-S)Pm,

ViPi,

a n d , for the case o f the linear c e n t r o i d finding filter A = P(m -

D=~Vi

l-S), ~(i

B = P(m - S) ,

- s) dS

References [1] E.. Gatti, A. Longoni, H. Okuno, and P. Semenza, Nucl. Instr. and Meth. 163 (1979) 83. [2] E. Gatli, H. Okuno, and M. Artuso, Nucl. Instr. and Meth. 167 (1979) 417. [3] J.L. Alberi and V. Radeka, IEEF, Trans. Nucl. Sci. 23 (1976) 251. [4] V. Radeka and R.A. Boie, IEEE Trans. Nucl. Sci. NS-27 (1980) 351. [5] W. Shockley, J.A. Copeland and R.P. James, in: Quantum theory of atoms, molecules and lhe solid state (Academic Press, New York, 1966) p. 537.