A square root metric in the analysis of the measurements of high energy physics instrumentation

Nuclear Instruments and Methods 200 (1982) 369-376 North-Holland Publishing Company

A SQUARE ROOT METRIC IN THE ANALYSIS ENERGY PHYSICS INSTRUMENTATION L. P E T R I L L O

369

OF THE MEASUREMENTS

OF HIGH

a n d M. S E V E R I

Istituto dt Fisica G. Marconi, Universita di Roma, I.N.F.N, - Sezione di Roma, Italy

Received 2 July 1981

A vast category of detectors are characterized by a charge output with a distribution which has a variance proportional to the mean value. In all these cases a square root scale achieved in the hardware seems more suitable from the point of view of the resolution of the measurements, of the number of ADC channels needed, and of a preliminary analysis in the stage of tuning and checking of the detector.

1. Introduction

(l=f~#=khE

For a detector which uses a photomultiplier there are: E the energy converted in photons in the sensitive part of the detector; rl the number of electrons produced in the photocathode; q the charge collected at the anode and applied to an ADC. The form of the functional dependence between these quantities

q--q(•) depends on the characteristics of the structure and of the operation of the whole detector. In the case of linearity where, among other things, there should be no saturation these relationships become

and among the variances

o2(~) ~ o2(h) + o 2 ( e ) ,~2

~2

o'(q) ~ ( k ) +°~('7) ~2 ~2 C_/2 It is known that the multiplication of the chain of dynodes also produces a continuity effect in the sense that it transforms the discrete variable ~1 into the variable q which can be considered continuous to a fairly good approximation. Should it be possible to disregard any fluctuation in this multiplication phase, and to consider the fluctuation in the multiplication to be confined to between the first pair of dynodes {11, this is to say:

~q = h E

o2(k) <
q = krl = k h E ,

one obtains

where h is independent of E and takes account of the conversion of E to photons, of the efficiency of collection them up to the photocathode, and of the efficiency of the cathode to convert them into electrons; k is independent of rl and takes account of the multiplication of the chain of dynodes (eventually dependent for its part on over-rating) and of the efficiency of the transmission to the A D C of the charge collected at the anode. Under these conditions of independence among those which have more correctly to be considered random variables, they are valid, apart from the relations among their mean values ~l=hE

0167-5087/82/0000-0000/$02.75

~) 1982 North-Holland

E- 2

o2(q) 02(rl)

~2

(l)

°2(h)-t

~2

~2

°2(E)

(2)

~,2

This relation shows that the relative fluctuation of q is determined just as much by the intrinsic relative fluctuation of E , o ( E ) / ~ L , as by the relative variability, o ( h ) / / ~ , of the phenomena which determine/7, related, on its part, to the characteristics of the detector. On the other hand q, with its distribution, is the only experimental data which can be used to obtain the physical data of interest i.e. the value of E and eventually its distribution, for which one has o,..(E)

o(q)

E

q

(3)

L. Petrdlo, M. Severi / High energy physics instrumentation

370

In order to obtain the value of o(E)/ff~ from eq. (2) it is necessary to know the instrumental constant o( h )//~ which can be obtained with an independent measurement consisting of exciting the detector with events different from those being examined, that is to say, with events for which it is known a priori that

o(- E <)

-

<

o(- h )

(4)

o2(r/) =F/.

The distribution of q, for what has already been said, is a distribution of a continuous variable and has the same set of moments as a Poisson distribution: as it is known, this is sufficient to characterize it completely, but for a scaling factor. Functions of this type will be called G L functions. From (2), (3) and (4) one then has

o(q)

/7

#

which becomes .---

~iEt,

Om,s( Et ) -having put

Often the distribution of 7/can be considered Poissonian; it occurs when the emission of each electron from the photocathode is independent of the emission of the others and happens with a probability which is small, being the product of the probabilities relative to the elementary physical processes which have contributed. In such a case, in particular, one has

Orals(E)

of the second equation of (6) is obtained once again, i.e.

o(~l)__ I ~

a2E0_ 1 + y g

'

where ~, is the ratio between the non-Poissonian contribution to o~i~(E t) and the Poissonian one. One obtains also omi~(E,)

o(q)

o(rl)

/l+y-

l

(S)

having defined the new parameter rl P-l+y which appears to be more significant because it establishes a complete analogy between (8) and (5) which also results from the relations that come from (8) o ( q ) - - ¢ ( 1 + "y) k-v/~

(5)

Vr~- '

o(',1) - I ~ - ~ V ~ -

(9)

from which

o(q)--~ocvC~ °.i,(E) - v'E

~/~.

(6)

This situation occurs, for example, in Cherenkov counters or in scintillators crossed by an ionizing particle. For other detectors, such as sandwiches of scintill a t o r - P b or lead glass arrays, the situation is more complicated because of a considerable variability in the development of the e.m. shower in the detector or of its distribution in the single elements of the same. This variability can determine on the distribution of r~, and therefore of q, an effect not to be ignored with respect to the simple Poissonian type to which, however, it is superimposed. Although this latter variability is connected with the structure of each detector, e.g. to its geometry, it is experimentally obtained, when the total energy E t of the particle is released in the detector: °mis(Et) - A +

(7)

where E t is measured in GeV, E 0 = 1 GeV, and where for A and a typically one gets A 2 6 . 5 × 10 -3 and a -~ 0.05-0.15 depending on the type of detector. If in eq. (7) A can be neglected, a relation of the type

Lastly, it has to be observed that from the first of eqs. (9) a measure of ~,, which could be of great interest, is connected to a possible independent measurement of k.

2. Measurement scales Normally a charge signal is analyzed by an A D C characterized by its maximum number of channels, c h m ~ , and by its error of sensitivity, that is by the width of a single channel, Aq, which produces the effect of discretizing the measurement, ch(q), of q. If (8) is valid and therefore the first of (9), it follows that: a) acting on k it is possible to do it in such a way that at a given point of the scale, corresponding to a given value of ~ and therefore of k~, we obtain the desired relative resolution, i.e. the prefixed value of

8(q)=

~-

Aq

--V~°:'V~"

(10)

It should be noted that in a zone in which 8(q) is to() large, the A D C will have a uselessly small Aq, which will produce a too large chin, G on the other hand in a zone in which 6(q) is too small it will be impossible to reveal the characteristics of the distribution of q - in the

L. Petrillo, M. Severi ,/ High energy physics instrumentation

371

statistical fluctuation of the estimate of o is comparable with that relative difference, that it is

o*-o o(o)~

0%0

o

0

0.1

0

!

I

I

1

2

3

I

~

ID

4

Fig. 1. Plot of ( a * - o)/a vs. 8 as discussed in section 2.

extreme case, they will be lost completely when the whole distribution is contained in a single interval. Even before this limiting case, a small enough value of 8(q) can produce on the estimates of ~/and o [2] systematic effects which can impair the information connected with those quantities which increases with the number of used events [see, for example, eqs. (19) and (21)]. As an example in fig. 1 the quantity (o* - o ) / o is shown vs. 8, o being the standard deviation of a Gaussian, and o* the one calculated when the same Gaussian is discretized on intervals of width A = o / 8 . This systematic effect can be easily corrected and it is worth doing when the number of events is at least comparable to the number, n*, of events so much so that the relative

o

1 f2(.*-

1)

In fig. 2 the behaviour of n* vs. 3 is shown. b) the total number of events being equal, experimental distributions with different ~ have an average number of events, n', contained in the modal channel proportional to (~/) - ] / 2 something which must eventually be born in mind in the case of a storage A D C . This situation is shown in figs. 3, 6 and 8 in which groups of five Gaussians characterized by o2(q)o:~l are shown (these are the functions to which the G L functions tend for large values of ~) and in fig. 10 which shows a group of three Poissonians. These disadvantages could be avoided by using a different metric, which can be obtained with a change of scale that contracts the scale at higher values of q and dilates it at lower ones. A logarithmic scale, suitable in other cases [3], has an excessive effect when conditions (9) are valid, as we can see in fig. 5. For that previously mentioned the metric

y="v~

(ll)

seems to be the most suitable because from the first order relation o2(y) ~ (dyl2

~q lOa2(q)

we obtain a2 o2(y) ~-(i

+ y ) k = const.

(12)

which means that n'(y), o ( y ) and 8 ( y ) = o ( y ) / A y are independent of the value of)7 and therefore of q and ~.

IO'n*

2

lO

..........A ............A,...........A .............A...,

lo'

500 I

i

I

1

2

3

q

I000

I

d

4

Fig. 2. Behaviour of n* with variation of 8 as discussed in section 2.

Fig. 3. With arbitrary units, five Gaussians, g(q), are shown which are characterized by the same surface and by q =49, 196, 400, 625, 900; o =3.5, 7, 10, 12.5, 15; ~ = 196, 784, 1600, 2500, 3600 from which one gets (I + "r)k =0.25 u.m.

L. Petrillo, M. Severi / High energy physics instrumentation

372

~l.J! F

0 . . . . ,

. i ......

. . . . .

i..

1000

.t y

5 0

¥

500

ij

1000

Fig. 7. The same as fig. 4 with reference to the functions of fig. 6.

Fig. 4. With arbitrary units, the functions (2/32).v/q .g(q), obtained from those of fig. 3 with the change of scale y = 32v~, are shown.

A 5OO

........................i!

..... . . . . . .

500

o

lOOO

V

q

1000

Fig. 8. With arbitrary units five Gaussians are shown which are characterized by the same surface and by q =49, 196, 400, 625, 900; o = 14, 28, nO, 50, 60; ~ = 12.25, 49, 100, 156.25, 225 from which one gets ( I + y) k = 4 u.m.

Fig. 5. With arbitrary units the functions (q/145).g(q), obtained from those of fig. 6 with the change of scaley = 145 In q, are shown.

.......l 0

......A 5

....... q

1000

Fig. 6. With arbitrary units five Gaussians are shown which are characterized by the same surface and by q =49, 196, 400, 625, 900; o =7, 14, 20, 25, 30; f =49, 196, 400, 625, 900 from which one gets (1 + y ) k = 1 u.m.

U

500

y

d_._... • ! 1

10o0

Fig. 9. The same as fig. 4 with reference to the functions of fig. 8.

L. Petrillo, M. Severi / High energyphysics instrumentation

373

3. Further discussion of a square root scale

The distribution of y, ~ ( y ) , is obtained from the relation ~(y) dy =/(q) dq from which •

¢

¢p(y)=f(q)~),=2

•

¢

c~f(q).

•

It obviously depends on the actual form of (9). In particular one has ¢

~_" . •. . . . . . . . . .

•

"'..

2~:. . . . .

"~ • . . . . . . . . . . . . . . . . . . ,_. . . . . . . . . 1

,

L'

)7 = fy~p(y) dy = afvCqf(q ) dq 0.2(y) = fy2ep(y) d y = a27/-)72.

.

However, from the Taylor series of (11) one gets Fig. 10. With arbitrary units the Pois¢onians P,,,(x) with m = 5, 10,20 are shown.

1 P'3(q) -t 16 ?/3

l o2(q)

)7~av/~

1

8

?/2

a2(0.2_(q) o2(Y)~-4 q This situation is shown quite clearly in figs. 4, 7, 9 and !1. An immediate consequence of (12) and of the fact that in first approximation

)7--a¢~ )7

[

"

y~a~/-q I _)7-0

~

o(y) -0.(y)

21/~-

= 2¢r~-

v~-f+ ~)k

(14)

1 #3(q)5 2 ?12 +

8 ~

=aC'~[l-Rc

",

128 ~2 t - . . .

o2(y)~T(l+y)k O

0.(y) ~ ~ [ I

= o¢~[1

[

+ RG..~s(0.)]

8 - ~ - 12---8~2

G

12~8 ~3 . . . .

- Ro~()7)]

0.(y)~i~[l

,t4*e

*

]

151+...) I + ~-~ ~-

0.~(y)~-a-(l+y)k 1+ ~ 4

) "'"

2) i f f ( q ) is a G L function, then y ~ av~- 1

¢

bt,l(q) q3 +

.... ()7)]

a2

which, when compared to (7), shows that, keeping equal, with a square root scale one obtains a resolution with respect to the zero of the scale which is twice that of a linear scale.

16

P,4(q) ) ?/4 + "'"

from which, for example, the two particular cases are made explicit 1) i f f ( q ) is Gaussian with o2(q) c~ ?/, then

(13)

is that

5

128

~-l + ~ 6 ~ +

...

+Rod0.)].

#

e

The four R functions are graphed vs. ~ in fig. 12. Finally, it is known that the skewness @

=/3.3/0

•

, " "

'~l~ . . . . .

i~E-..*t

.....

~ t ....

* ?..t

1

10

• . . . .

• I ~---

ee t ....

J..*t~..t

l

~0

30

x'

Fig. 11. With arbitrary units the functions obtained from those of fig. 10 with the change of scale x'--6Cx- are shown.

.3

is =-- 0 for a Gaussian, while is equal to ( ~ ) - 1/2 for a G L function on a linear scale. Conversion to a square root scale tends to decrease in fl(y) the positive contributions and to increase the negative ones with the net result shown in fig. 13. Now, ignoring the quantities R, that is considering (14) valid, the number of channels with which )7 is

374

L. Petrillo, M. Seven / High energy physics instrumentation

o

-1

10

~.

- ....

•,, \ . _ ,,, N

-- -- -- R Gt.(o)

R~,=do) ch

~L''\

162

~

RGL(V ) \ "\

N~\\" \ \ "\

1~ 3.

to'.

"%., N~\ \ . .

10 4 !

u

101

I

10 2

,8

ID

T

10 3

I

I

I

101

102

103

I

"

104

Fig 12. Four R functions vs. ~ as discussed in section 3.

Fig. 14. Number of channels and number of bits vs. =, parametrized by 8, as discussed in section 3.

measured is

the maximum value of the error, that is /)max ~ ~max

"o(y) and in particular, from (l), ch m~( Y ) = ch( .V~.~ ) -- 2 8, ~V/-~'~

(l 5)

where 8). = const, is to be chosen a priori based also on what has been said in point a) of sect. 2. This relation of practical importance is graphed in fig. 14, from which one gets, for example, that. having put 8y = l for which it is o* - o / o "=--3%, for ~'m,~= 103 only 6 bits are necessary, while for Vm,, = 104 8 bits are necessary. When p,~,, is known, it could be interesting to know ~m~,, i.e. the maximum mean value of a distribution of v fully contained within the scale. This can be obtained from the known convention of thinking of 30 as, in a sense,

-~-3Om.~(~') = Vm.x-~-3~ma:

the behaviour of which is shown in fig. 15. Considerations of the same type for a linear scale are not so immediate because 8(q) is not constant. Having found ~' for which 8(q) has the chosen value and having put c-/' = c qm~x with 0 < c ,~ 1, one derives q,

chm~(q)

~ Aq

_

1

c

~(o') L'

1 8(,~,)(~

o'--~-

= 8(#') ~- ~-m~ -- 8(#') ~-( ~'~r.a~+ 3 ~ (16) From(15) and (16) one obtains the ratio between c h ~ ( q ) and c h , . ~ ( y ) vs. ( when 8,. = 8(0') and keep-

_] (~.),

\ N

"--

\

-~,us.

\

103.

\\ ~\-\ 10 ~,

\x\

"~\.\

18. \X\\~'\ x\\

10 2

10-

NNN~\

//////

,

,

,

,

101

10 2

10 3

10 4

~'~

// 10

Fig. 13. Behaviour of the skewness in the Gaussian and GL cases vs. ~.

10 2

10 3

v~..

Fig. 15. Behaviour of ~,.,a,, vs. vm,,,,, as discussed in section 3.

375

L Petrillo, M Severi / High energy physics instrumentation

1

2~7

l o o.

-|

lO

lb 3

10 ~

16'

lo0

Fig. 16. Behaviour of r(c) vs. c as discussed in section 3.

ing

lPmax

equal

chm~(q) 1 r ( , ) -- ~chmax(y - 2~-~ '

(17)

From this simple relation, graphed in fig. 16, one gets, for instance, that it is sufficient to put c ~ 0.25 to obtain r(~) > 1.

4. Applications To obtain the characteristics (12), (14), (17) of a square root metric it is necessary either that the change of scale (11) is carried out with an analog circuit the output of which is then applied to a linear A D C or, equivalently, that the quantity q is applied directly to an A D C with a conversion ramp with a square root behaviour * To carry out (I 1), instead, in software starting from a linear A D C does not realize the constancy of the relative resolution which is one of the most significant characteristics in the case of measurement of a single physical event. In fact, when carrying out a single measurement, 8 is the significant quantity to judge how well the instrument used is suitable, since it is the ratio between the intrinsic uncertainty of the measured quantity and the one of the instrument. To have the possibility of using a measurement system with 8 : const, means obtaining measurements all carrying the same amount of information. As it was seen, this is realized automati-

* The LeCroy ADC mod. 2250 Q has a conversion, on 512 ch, of the q = (0.54× 10-3 ch2 +0.24 ch-2.45)pC which, however, does not satisfy the metric characteristics mentioned above.

cally with a square root metric provided that the distribution of the quantity to be measured satisfies (9). In practice this fact produces two complementary effects: it improves the quality of the measurements at the lower values of the scale, and eliminates the waste of relative sensitivity at the higher ones, therefore allowing the number of A D C channels to be reduced to those really necessary, once it has been possible to value Pmax, with possibly considerable saving in hardware. Obviously, the wider the dynamic range of the measurements that the detector has to carry out, the more important is all this. It must be noticed, finally, that the necessary return to a linear metric via software for further analysis of the measurements does not change these considerations. Even in the case of spectra of measurements the square root scale can be useful because in many cases it allows a more rapid analysis of them, above all in the tuning or controlling stage of the detector. For example from (12) it follows that it is easier to make evident and value possible dependences of y on parameters such as the entrance point of the particle into the detector, or as the actual value of the measure itself, i.e. a dependence of the type y = y(E). A case where it is necessary to examine the outputs of the PM altogether and not as single events is also that in which one wants to carry out the control of the gain of the PM, or more clearly, of the parameter k. Normally the control is done by analysing the outputs of the PM produced by light pulses generated by a long-term-stabilized pulse generator and one that causes an emission of photoelectrons comparable in number to that of the physical events. Even in this case (9) is valid where y now represents the non-Poissonian contribution to the fluctuation of the light pulses produced. A first type of evaluation of the stability of k can be obtained starting from the possible variations of the arithmetic mean of a certain number of outputs: having called 00 and qt those means at a reference time and at a generic time, respectively, the following quantity is obtained p ( t ) -- kt -- k ° -- qt -- qo

k0

00

(18)

from which, when p is small enough, o(q)

o(0)

q

/I

1

V"n0+--nt

(19)

is obtained, where n o and n t are the number of light pulses used at the reference and the generic time. Of course (18) and (19) are valid only in the case of a good long term stability of the pulse generator, i.e. of y and of ~. While it is possible to obtain generators with negligible values of ~, [4,5], as far as the constancy of ~ is concerned the possible alternatives of using generators which are fairly stable but expensive, (something that

L. Petrillo, M. Severi / High energy physics instrumentanon

376

can be prohibitive, particularly when the n u m b e r of P M s to be controlled is high) or of using less sophisticated and therefore more economical ones which, however, can have a less long term stability [6]. T o overcome this inconvenience one can use a more complex analysis of the same data, such as c o m p u t i n g

o 2 ( q ) / ~ ~ (1 + r ) k

(20)

that, when y is k n o w n or recognized to be negligible, gives a direct measure of k which is i n d e p e n d e n t of ~, that is of the efficiency of the light emission of the generator. F r o m (20) one gets

°2(k) k2

°2[s2(q)] + s2(q)

I(2 + I) ~ - 2

oZ(q)

-

?12

n

(21)

n"

n being the n u m b e r of pulses on which the estimate, s2(q), of the variance is obtained. The use of a square root metric, which for (12) is

characterized by a c o n s t a n t value of o2(y) on the whole scale, allows a valuation of k i n d e p e n d e n t of ~ even simpler than that of (20) and with an error still given by (21). Even a possible analysis on sight of spectra of events of this type is m a d e easier.

References [1] G.T.. Wright, J. Scient. Instr. 31 (1954) 377. [2] M. Severi, Introduzione alia Esperimentazione Fisica (Ed. C.I.S.U.) Par. A. 30. [31 S. Whittlestone, Nucl. Instr. and Meth. 169 (1980) 215. [4] A System for PM's Gain control, Internal Note I.F.U. of Rome No. 727 (1979). [51 Behaviour of Pulsed LEDs, Internal Note I.F.U. of Rome No. 745 (1979). [61 M. Chiaudano, Thesis for a degree in Physics, University of Rome (1978).