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Dependence of hadron calorimeter energy resolution on the properties of a wavelength shifter
L~V~L6
Nuclear Instruments and Methods in Physics Research A322 (1992) 207-210 North-Holland
,~1
1 V
si
L17
V (~l(~ i ~ Li~ca .-` ''.~
7
L~
Se (.t;or.A
epen ence of hadron calorimeter energy resolution on the properties of wavelength shifter v.s.
ulnyantsev
B.1. Stepanou histiture oj'Physics of Belarasian Acaclenty of Sciences, Minsk, Belarus
Received 26 February 1992 An analytical expression which describes the dependence of a hadron calorimeter energy resolution on the main properties of a wavelength shifter is presented. The above mentioned properties are the light attenuation. the length and the degree of light reflection at the free end of the wavelength shifter. The expression obtained is compared with experimental data . 1 . Introduction At present the hadron sampling calorimeters with active elements designed on the base of scintillating materials are actively built and exploited [1-4]. As the active elements either scintillator plates [1,2,4] or scintillating plastic fibers [3] are widely used . In the first case the light from the scintillator plates is collected and transferred to a light-sensitive device (e .g . a photomultiplier (PM)) by means of a special wavelength shifter (WLS). In the second case the scintillating plastic fibers carry out both the functions of the active elements and of the WLS simultaneously . The known disadvantage of such constructions is the non-uniformity of the light collection, which is caused by light attenuation in the WLS . For the reasons given, the amount the light, incident to the PM depends both on the shower "luminosity" and on the depth of the shower emergence inside the calorimeter. As a result, the distribution of the calorimeter signals, exited by the monoenergetic hadron beam, acquires the finite width, giving an additional contribution to the calorimeter energy resolution . At present the above mentioned contribution is described by the expression (1 - exp(-A,/A)) [5], where A, is nuclear interaction length, and A is light attenuation length in the WLS. The typical values are A t = 0.2 m, A >_ 1.5 m. However, this expression does not take into account many important features of the light propagation through the WLS and, as a matter of fact, it is only the first approximation . For the calorimeters of the given construction the influence of the WLS nonuniformity on the energy resolution is usually estimated by means of Monte îarlo simulations [6]. However, in a process of designing new calorimeters, it is an analytical expression . which describes in
greater depth the dependence of the energy resolution upon the; A,/A ratio, upon the WLS length, and upon the degree of the light reflection at the free end of the WLS, that will be of interest . The purpose of this paper is to obtain the analytical expression mentioned above.
2. Analytical expression Let A be the value of a signal, generated in a calorimeter by one hadron shower . Define the energy resolution as aA/(A ), where oA and (A) are the standard deviation and the mean of the A distribution, respectively . Let us use the following notations: 0(y) is the density of shower energy distribution along the longitudinal axis of the shower (the starting point of the shower lies at y = 0) ; L is the length of the WLS, equal to the length of the sensitive volume of the calorimeter ; a is the luminous reflectivity at the free end of the WLS. The light attenuation on a path length x in the WLS is described by exp(-x/A). The arrangement of the shower, of the WLS and the PM along the axis OX is shown in fig. 1 . We assume that the WLS collects the light from the calorimeter volume, the transverse dimension of which coincides with the one of the shower. Let us find the signal A(x), created by the shower emerged in the calorimeter at a depth x. Consider the light from the elementary transverse shower layer dy, located at a distance y from the starting point of the shower. Having entered the WLS, the light is divided into two equal parts. The first part goes to the PM (the path length is L - ( x + y)) at once . The second one first goes to the free end of the WLS and after being
0168-9002/92/$05 .00 ,O 1992 - Elsevier Science Publishers B.V . Ali rights reserved
208
VS. Rrrrrn~lrrrtsc~r
wovelength shifter
shcwe, QroFile
/ lladron ralorinrrter rrrergy resolution
PM
variable r, is it random variable itself. The nth moment of the f1 distribution is defined by the integral (A
I I
x o Fig. 1. A schematic view of the arrangement of the shower, the WLS and the Purl alone the axis OX .
reflected (a reflectivity) also comes to the PNI (the path length is L + (x + r)). We assume: that both contributions of the light arrive at the PM at the same time . Then the signal A(x) is the integral over the volume occupied by the shower in the calorimeter A(x)
=
1 in,
,1
exp (
+a exp -
L - (x +y) A
A
In eq . (1) and below we omit the negligible common coefficients which are reduced in the calculation of As a first approximation we suppose h(y) _ exp(- %./A I ) . This approximation is sufficiently close to reality. It is known that the longitudinal shower profile is well described by the dependence ! ." exp(- by ). The parameters ct and h depend logarithmically on the energy and at 1(1 GeV (for lead and uranium absorbers) they are equal to ct = 0.06, h = 0.9/A I [7] . If (b(O =exp(-t'/A I ) . eq . (1) is easily integrated . and using the dimensionless quantities t = x/A I ; 1= L/A, and 6=AI/A we obtain A(t)=f(t)-f(1) exp(t-1),
with
(r) =(1/h_) exp(6t)+a(l/b + ) exp(-St), and b+
=I+à,f(1)=f(t=1) .
sh®-lu!d be noted, that even at absolute transparency of the WLS (A - ou) the dependence of A upon t = x/A I does not vanish it
A(t)
A ---
(I +a)(I - exp(t-1)) .
This dependence is caused by shower leakage through the back edge of the calorimeter and leads to nonzero U,IlCA) The position x of the shower starting point is the random variable distributed according to /t(x) ( I /A I ) exp( -x/A I) in the range from 0 to L . Consequently, the signal A, being a function of the random
= f 1',4 ,l ( .t ) lt ( .r ) la
d.r .
In terms of the first- and second-order moments the energy resolution is expressed by -, (T .tl(A>=((A-')/( ,,1) -'- I)I, The considered integrals are exactly evaluated and we have ( .
and ~1
~b ( y ) d Y .
)
= exp( -1)
exp(M ) h-
exp( -1S ) ++If(1)I , b
(A-,)=]',--1 , . with _
1
2a
a-
-1,=exp(-1)R(S . 1, a), and R(S, 1, a) L'xp(21S)
2a
b 2 (I -26 ) + b+b- +
f(1)
exp(16) - 1 h_
a 2 exp( -216 ) h+(1 +26) +a
I -exp(-16)
,
- f (1)(l - exp( -1))
h+
The variables S, 1, the functions b +_ and f(1) were defined in eq . (2). The curves in figs . 2a, 2b and 2c illustrate the dependence of the energy resolution upon the attenuation length A (in units A/A I) for several values of L. Each figure presents the dependence both with the reficction (a = 0.8) and without the one (a = 0) at the free end of the WLS. The behaviour features of (T,,/(A) are most clearly seen from the curve with L - 6A I and a = 0.8 . With increasing A, the value of ff,I/( A) falls down rapidly, reaches the minimum value and then rises slowly to the limiting value. The latter corresponds to the absolute transparency of the WLS (A --> -) and is equal to (?,jA) -- Ir., exp(-L/AI) With al precision not lower than 2rß .
V.S. Rwnyant.wi" / Hadron calorimeter eiter*- resolution In fact, eq . (6) defines the energy resolution caused by the shower leakage through the back edge of the calorimeter. The other curves have the same features but they are revealed at values A greater than 25A, .
DA
4. Approximate expression In addition we present the approximate expression for the energy resolution o~-,l(A)= - 1,+2-1,f',- -1 ,If,, b2 b2-
a' +2,3) , b+ (1
, = .f(l)(1 +1) exp(-1), .ii = 1/b, +alb+ , ,,=
2f(l) exp(-1) S +a
exp(13) - 1 h
b
020 ~n=0
0 15
.-"B ,
0 10
015
1
000 ' . . 0
®-0
1
\a=0 8 005
_ . 5
10
15
i
. 20 A/A 25
. . 000 ~___ . ._ . . . . .___ . . . . 0 5 10 15 20 . \/AQ25
Fig. ?. The energy resolution as a function of the light attenuation length of the WLS (in units A /A `) when the P is located at the back end of WLS. (a). (b). and (c) stand for the lengths of WLSs (in units L /A,) equal to 6. 7. and t;. respectively. Here a is the luminous reflectivity at the free end of the WLS. A, is the nuclear interaction length . The dependence of Q.,/( A> upon A/A, is shown in figs. 3a, 3b and 3c for L/A, = 6, 7 and 8, respectively. In comparison with the version when the PM is located at the back end of the WLS, the following features can be observed . The values o~ .y ( A ;. when the parameter a = 0 and A/A, < 10, demonstrate a small improvement of the energy resolution . In all other cases it gets
ffA
(020 015
I - exp( -15 )
0.10
b+
iri the range of A/A, >- 5.5 and L > 6A, the difference between the exact eqs. (3)-(5) and this approximate expression does not exceed 8% . In the framework of the approach considered it is easy to evaluate the energy resolution in case, when the PM is located at the front end of the WLS. So, the signal value as a function of the shower depth inside the calorimeter is A(t)=g(t)-g(I) exp~t-1), 061 ~(rl- exp[(l -
,.A>
005 ~ v
Let us apply the obtained formulas for the description of the properties of the calorimeter built and tested by the SPACAL Collaboration [8]. In this calorimeter the scintillating fibers were used as an active material and as WLSs . The energy resolution of the calorimeter according to ref. [8] is a!_/E = 31 _0 1',~~ / E[GeVI + h, where h = 2.6% . As shown in ref. [9], the contribution of the light attenuation in the fibers to the term h amounts to I .VL On the basis of the parameters quoted in ref. [8] (a = 0 .85, L = 2.0 m; A, = 0.21 m, A - 3.5 m) and using eqs. (3)-(5) we obtain (T .,/(A) = 1 .7(' . This value is in close agreement with the experimental one of 1 .Vi, .
2a
aA
L /A . =E
020
010
3. Application to an experiment
O
'tl9
acxp[-(1-t)8]
+
-
g(1)=g(t=1), b + = I +5 . The variables h, t and 1 are the same as in eq . (2).
005 000
QA
_
(A )
0 20 0'5
4
010 ~
coo
. . . .
5
. . ..
0
.
. . 5. . . " 2 " % 2 0 \ ,\ 5
Fig. 3. Dependence similar to the one presented in fig. 2. but for a case . wher, the PM is located at the front end of the WLS . (a) . (b) and (c) are the same as in fig ._'.
210
VS. Runqantsev / Hadron calorimeter energy resohition
worse. The limiting values of ca-,/(A) if A unchanged and are defined by eq . (6).
remain
5. Conclusions
The presented calculations permit me to draw the inferences as follows: 1) Under other equal conditions, the better energy resolution is achieved . when the P is located at the back end of the WLS. 2) The reflecting front end of the WLS provides an essential improvement of the energy resolution (for example. when L = SA 1, A = 15A 1. the increase of from 0 to 0.8 leads to decrease of or-, A > by factor C, about 2) . 3) When the luminous reflectivity a 0.8. the minimal energy, resolution is achieved already at A = 15A, and the subsequent decrease of o,,&4> can be obtained only b~ increasing the longitudinal calorimeter size .
Acknowledgements I would like to thank Ju .A . Budagov, A.M . Zaitsev, I.E . Chirikov-Zorin, EN . Doktorov and T.P . Shaldina who promoted in one way or another the appearance of this paper. eferences [1] E. Bernardi et al ., Nucl . Insu . and Meth . A262 (1987) 229. [2] G.A . Alekseev et al ., Preprint IHEP, 90-157, Serpukhov (1990) . [31 D. Acosta et al ., Nucl. Insu. and Meth . A294 (1990) 193. [41 T. Akesson et al .. Nucl . Insu . and Meth . A_162 (1987) 243. 151 V.M . Buyanov et al ., Preprint IHEP. 89-45, Serpukhov (1989). [6] T. Akesson et al ., Nucl . Insu . and Meth . ANI (1985) 17 . [7] R.K. Bock et al ., Nucl . Insu. and Meth . 186 (1981) 533. [8] H.P . Paar, in : Proc. Syrup . on Detector Research and Development for the SSC. October, 1990. Fort Worth, Texas, USA. eds. T. Dombeck, V. Kelly and G.P . Yost (World Scientific) p. 391 . R. Wigmans. 1991, preprint CERN-PPE/91-39 . CERN, Geneva .
Nuclear Instruments and Methods in Physics Research A322 (1992) 207-210 North-Holland
,~1
1 V
si
L17
V (~l(~ i ~ Li~ca .-` ''.~
7
L~
Se (.t;or.A
epen ence of hadron calorimeter energy resolution on the properties of wavelength shifter v.s.
ulnyantsev
B.1. Stepanou histiture oj'Physics of Belarasian Acaclenty of Sciences, Minsk, Belarus
Received 26 February 1992 An analytical expression which describes the dependence of a hadron calorimeter energy resolution on the main properties of a wavelength shifter is presented. The above mentioned properties are the light attenuation. the length and the degree of light reflection at the free end of the wavelength shifter. The expression obtained is compared with experimental data . 1 . Introduction At present the hadron sampling calorimeters with active elements designed on the base of scintillating materials are actively built and exploited [1-4]. As the active elements either scintillator plates [1,2,4] or scintillating plastic fibers [3] are widely used . In the first case the light from the scintillator plates is collected and transferred to a light-sensitive device (e .g . a photomultiplier (PM)) by means of a special wavelength shifter (WLS). In the second case the scintillating plastic fibers carry out both the functions of the active elements and of the WLS simultaneously . The known disadvantage of such constructions is the non-uniformity of the light collection, which is caused by light attenuation in the WLS . For the reasons given, the amount the light, incident to the PM depends both on the shower "luminosity" and on the depth of the shower emergence inside the calorimeter. As a result, the distribution of the calorimeter signals, exited by the monoenergetic hadron beam, acquires the finite width, giving an additional contribution to the calorimeter energy resolution . At present the above mentioned contribution is described by the expression (1 - exp(-A,/A)) [5], where A, is nuclear interaction length, and A is light attenuation length in the WLS. The typical values are A t = 0.2 m, A >_ 1.5 m. However, this expression does not take into account many important features of the light propagation through the WLS and, as a matter of fact, it is only the first approximation . For the calorimeters of the given construction the influence of the WLS nonuniformity on the energy resolution is usually estimated by means of Monte îarlo simulations [6]. However, in a process of designing new calorimeters, it is an analytical expression . which describes in
greater depth the dependence of the energy resolution upon the; A,/A ratio, upon the WLS length, and upon the degree of the light reflection at the free end of the WLS, that will be of interest . The purpose of this paper is to obtain the analytical expression mentioned above.
2. Analytical expression Let A be the value of a signal, generated in a calorimeter by one hadron shower . Define the energy resolution as aA/(A ), where oA and (A) are the standard deviation and the mean of the A distribution, respectively . Let us use the following notations: 0(y) is the density of shower energy distribution along the longitudinal axis of the shower (the starting point of the shower lies at y = 0) ; L is the length of the WLS, equal to the length of the sensitive volume of the calorimeter ; a is the luminous reflectivity at the free end of the WLS. The light attenuation on a path length x in the WLS is described by exp(-x/A). The arrangement of the shower, of the WLS and the PM along the axis OX is shown in fig. 1 . We assume that the WLS collects the light from the calorimeter volume, the transverse dimension of which coincides with the one of the shower. Let us find the signal A(x), created by the shower emerged in the calorimeter at a depth x. Consider the light from the elementary transverse shower layer dy, located at a distance y from the starting point of the shower. Having entered the WLS, the light is divided into two equal parts. The first part goes to the PM (the path length is L - ( x + y)) at once . The second one first goes to the free end of the WLS and after being
0168-9002/92/$05 .00 ,O 1992 - Elsevier Science Publishers B.V . Ali rights reserved
208
VS. Rrrrrn~lrrrtsc~r
wovelength shifter
shcwe, QroFile
/ lladron ralorinrrter rrrergy resolution
PM
variable r, is it random variable itself. The nth moment of the f1 distribution is defined by the integral (A
I I
x o Fig. 1. A schematic view of the arrangement of the shower, the WLS and the Purl alone the axis OX .
reflected (a reflectivity) also comes to the PNI (the path length is L + (x + r)). We assume: that both contributions of the light arrive at the PM at the same time . Then the signal A(x) is the integral over the volume occupied by the shower in the calorimeter A(x)
=
1 in,
,1
exp (
+a exp -
L - (x +y) A
A
In eq . (1) and below we omit the negligible common coefficients which are reduced in the calculation of As a first approximation we suppose h(y) _ exp(- %./A I ) . This approximation is sufficiently close to reality. It is known that the longitudinal shower profile is well described by the dependence ! ." exp(- by ). The parameters ct and h depend logarithmically on the energy and at 1(1 GeV (for lead and uranium absorbers) they are equal to ct = 0.06, h = 0.9/A I [7] . If (b(O =exp(-t'/A I ) . eq . (1) is easily integrated . and using the dimensionless quantities t = x/A I ; 1= L/A, and 6=AI/A we obtain A(t)=f(t)-f(1) exp(t-1),
with
(r) =(1/h_) exp(6t)+a(l/b + ) exp(-St), and b+
=I+à,f(1)=f(t=1) .
sh®-lu!d be noted, that even at absolute transparency of the WLS (A - ou) the dependence of A upon t = x/A I does not vanish it
A(t)
A ---
(I +a)(I - exp(t-1)) .
This dependence is caused by shower leakage through the back edge of the calorimeter and leads to nonzero U,IlCA) The position x of the shower starting point is the random variable distributed according to /t(x) ( I /A I ) exp( -x/A I) in the range from 0 to L . Consequently, the signal A, being a function of the random
= f 1',4 ,l ( .t ) lt ( .r ) la
d.r .
In terms of the first- and second-order moments the energy resolution is expressed by -, (T .tl(A>=((A-')/( ,,1) -'- I)I, The considered integrals are exactly evaluated and we have ( .
and ~1
~b ( y ) d Y .
)
= exp( -1)
exp(M ) h-
exp( -1S ) ++If(1)I , b
(A-,)=]',--1 , . with _
1
2a
a-
-1,=exp(-1)R(S . 1, a), and R(S, 1, a) L'xp(21S)
2a
b 2 (I -26 ) + b+b- +
f(1)
exp(16) - 1 h_
a 2 exp( -216 ) h+(1 +26) +a
I -exp(-16)
,
- f (1)(l - exp( -1))
h+
The variables S, 1, the functions b +_ and f(1) were defined in eq . (2). The curves in figs . 2a, 2b and 2c illustrate the dependence of the energy resolution upon the attenuation length A (in units A/A I) for several values of L. Each figure presents the dependence both with the reficction (a = 0.8) and without the one (a = 0) at the free end of the WLS. The behaviour features of (T,,/(A) are most clearly seen from the curve with L - 6A I and a = 0.8 . With increasing A, the value of ff,I/( A) falls down rapidly, reaches the minimum value and then rises slowly to the limiting value. The latter corresponds to the absolute transparency of the WLS (A --> -) and is equal to (?,jA) -- Ir., exp(-L/AI) With al precision not lower than 2rß .
V.S. Rwnyant.wi" / Hadron calorimeter eiter*- resolution In fact, eq . (6) defines the energy resolution caused by the shower leakage through the back edge of the calorimeter. The other curves have the same features but they are revealed at values A greater than 25A, .
DA
4. Approximate expression In addition we present the approximate expression for the energy resolution o~-,l(A)= - 1,+2-1,f',- -1 ,If,, b2 b2-
a' +2,3) , b+ (1
, = .f(l)(1 +1) exp(-1), .ii = 1/b, +alb+ , ,,=
2f(l) exp(-1) S +a
exp(13) - 1 h
b
020 ~n=0
0 15
.-"B ,
0 10
015
1
000 ' . . 0
®-0
1
\a=0 8 005
_ . 5
10
15
i
. 20 A/A 25
. . 000 ~___ . ._ . . . . .___ . . . . 0 5 10 15 20 . \/AQ25
Fig. ?. The energy resolution as a function of the light attenuation length of the WLS (in units A /A `) when the P is located at the back end of WLS. (a). (b). and (c) stand for the lengths of WLSs (in units L /A,) equal to 6. 7. and t;. respectively. Here a is the luminous reflectivity at the free end of the WLS. A, is the nuclear interaction length . The dependence of Q.,/( A> upon A/A, is shown in figs. 3a, 3b and 3c for L/A, = 6, 7 and 8, respectively. In comparison with the version when the PM is located at the back end of the WLS, the following features can be observed . The values o~ .y ( A ;. when the parameter a = 0 and A/A, < 10, demonstrate a small improvement of the energy resolution . In all other cases it gets
ffA
(020 015
I - exp( -15 )
0.10
b+
iri the range of A/A, >- 5.5 and L > 6A, the difference between the exact eqs. (3)-(5) and this approximate expression does not exceed 8% . In the framework of the approach considered it is easy to evaluate the energy resolution in case, when the PM is located at the front end of the WLS. So, the signal value as a function of the shower depth inside the calorimeter is A(t)=g(t)-g(I) exp~t-1), 061 ~(rl- exp[(l -
,.A>
005 ~ v
Let us apply the obtained formulas for the description of the properties of the calorimeter built and tested by the SPACAL Collaboration [8]. In this calorimeter the scintillating fibers were used as an active material and as WLSs . The energy resolution of the calorimeter according to ref. [8] is a!_/E = 31 _0 1',~~ / E[GeVI + h, where h = 2.6% . As shown in ref. [9], the contribution of the light attenuation in the fibers to the term h amounts to I .VL On the basis of the parameters quoted in ref. [8] (a = 0 .85, L = 2.0 m; A, = 0.21 m, A - 3.5 m) and using eqs. (3)-(5) we obtain (T .,/(A) = 1 .7(' . This value is in close agreement with the experimental one of 1 .Vi, .
2a
aA
L /A . =E
020
010
3. Application to an experiment
O
'tl9
acxp[-(1-t)8]
+
-
g(1)=g(t=1), b + = I +5 . The variables h, t and 1 are the same as in eq . (2).
005 000
QA
_
(A )
0 20 0'5
4
010 ~
coo
. . . .
5
. . ..
0
.
. . 5. . . " 2 " % 2 0 \ ,\ 5
Fig. 3. Dependence similar to the one presented in fig. 2. but for a case . wher, the PM is located at the front end of the WLS . (a) . (b) and (c) are the same as in fig ._'.
210
VS. Runqantsev / Hadron calorimeter energy resohition
worse. The limiting values of ca-,/(A) if A unchanged and are defined by eq . (6).
remain
5. Conclusions
The presented calculations permit me to draw the inferences as follows: 1) Under other equal conditions, the better energy resolution is achieved . when the P is located at the back end of the WLS. 2) The reflecting front end of the WLS provides an essential improvement of the energy resolution (for example. when L = SA 1, A = 15A 1. the increase of from 0 to 0.8 leads to decrease of or-, A > by factor C, about 2) . 3) When the luminous reflectivity a 0.8. the minimal energy, resolution is achieved already at A = 15A, and the subsequent decrease of o,,&4> can be obtained only b~ increasing the longitudinal calorimeter size .
Acknowledgements I would like to thank Ju .A . Budagov, A.M . Zaitsev, I.E . Chirikov-Zorin, EN . Doktorov and T.P . Shaldina who promoted in one way or another the appearance of this paper. eferences [1] E. Bernardi et al ., Nucl . Insu . and Meth . A262 (1987) 229. [2] G.A . Alekseev et al ., Preprint IHEP, 90-157, Serpukhov (1990) . [31 D. Acosta et al ., Nucl. Insu. and Meth . A294 (1990) 193. [41 T. Akesson et al .. Nucl . Insu . and Meth . A_162 (1987) 243. 151 V.M . Buyanov et al ., Preprint IHEP. 89-45, Serpukhov (1989). [6] T. Akesson et al ., Nucl . Insu . and Meth . ANI (1985) 17 . [7] R.K. Bock et al ., Nucl . Insu. and Meth . 186 (1981) 533. [8] H.P . Paar, in : Proc. Syrup . on Detector Research and Development for the SSC. October, 1990. Fort Worth, Texas, USA. eds. T. Dombeck, V. Kelly and G.P . Yost (World Scientific) p. 391 . R. Wigmans. 1991, preprint CERN-PPE/91-39 . CERN, Geneva .