ОбЪВАМНIННЫI ИНСТИТУТ

RABPHЫI ИCCIBAOBIHMI

АУ б на

El-84-418

B.Stowiriski, W .Czaj,* L.S.Okhrimenko, A.Bancerek~ E.Mulas, B.Redlicki; R. Wisniewski •

PHENQMENOLOGICAL DESCRIPTION

OF ТНЕ SPATIAL DISTRIBUTION

OF IONIZA.TION LOSSES

IN ELECTROMAGNETIC

SHOWERS PRODUCED ВУ GAMMA-QUANTA

WITH ТНЕ ENERGY Е1= 60-ЗООО . МеV

Submitted to XXII International Conference on High Energy Physics. Leipzig. July 19-25. 1984.

*Physical lnstitute of the Warsaw Technical University .

1984

I. X.ВODUCIUOJr

Ia 8Pit8 .o.t taa.e hot t!lat aU ele..n~ рJм-- 88UDs up а 088'0848 elecao.p"U.o Р""• (C:U) pжoituoetl 1W JdC1a ~ (1 »аа.,о1> .........-- .or .1 .. va, . ее .. u aawa·loaa accl11, t1a8n 18 ~ ·!d."'-'И .а1о1•~ -..ot 8114 оовrацаt. Ь8 t11e praoU.oal· IJ(Wat ~ n.w. letta'ip~oa ot . *• pa.u. _. .._.. ~oal "" ..... u.~ ~ th8 ...... . tia8a7· ...... ~' u . ...U.U to-~· 1D Up. ~. iR WldJdl ..... lialU:a.le 1:R1zlc ot .._ elaouou 18 -~• . ..ii laeU7 ... IIOI'Мat .. ·!а ..... ·~ ot". aD" --·ot tМ. ~· .~ ~ аьаn ~ ........ . ,.~ мr:l.oal n.tt • ..... ~· . ...: ~ "· ··~·.;.:г.с.'Оi

nctUU'e• а le~ ot · м1DII Qt .... to ь.tw. at 1Wt ..... i .. ;,··~IJ "'fWiiti ...... (~ •. :ra ·~~·~Z'!j·~·!iatii\~ ~:~~·~ . ... ~. ,.'-ti

pcmd1 .. ~ ot·· .... . ~. praou ...

Ia w.. ~pon ...,.о ~-· ·~ tЬ·..-. ".eirwl вr • 6о - эОоо ._, . а1111 tw cm...# :aiмNer .le~tz.oa ·.-.J ....ьм•• -о ..; О - 1. 5 J.Тef ..t .Э- м an pt88Jate4.

J • ' ,...., ! ... "8:,j

2. МЕТНОD AND EXPERIМENTAL DATA

The f ormulae prescnt ed in thi s report, descr ibing longitudi­nal and transversal SEIL distributions, have been obtained as а re ­sult of statistical analysis of the sample consisting of 2045 event в

of ES registered on pictures of the ХВС: the ХВС of JINR (Dubna) exposed in the beam of ~+ mesons at 2.34 GeV/c and the ХВС of IETP (Moscow) irradiated in the jr- meson beam at 3.5 GeV/c. In each selected event total shower electron ranges,д~ ri (Er,E

0,t,p)/

Llt.Ap, observed inside а square of the вurface ~ t др have been measurad. Vertices of such а вquare have coordinates : (t, t-At) -along the shower development axis (DSA), counting out from а pri­

mary ganuna-quanta conversion point, and (р, р- Ll р) - in t he direc­

tion perpendicular to а DSA. Here ~t = 2•др = О.588 • х0 , where

Х0 = 40.5 ± 1.7 mm/7/ is а radiation unit of liquid xenon. Later

in the text all lengths are expressed in Х0 units. Events of the sample have been devided among 23 energy intervals (ТаЬlе 1).

ТаЬlе 1

Numbers Nr of eventв of ES and corresponding values of the ener6Y - ~~ ь~·~~ ~.I..I.V~ • ..., ........ ...,.ul;!.. -6.1..,

нr Er (I.ieV) нr Er сыеv) 23 65 + 5 75 555 + 55 23 75 + 5 82 680 + 70 46 85 + 5 129 875 + 125

130 100 + 10 108 1125 + 125 143 120 + 10 92 1375 + 125 208 145 + 15 61 1625 + 125 190 175 + 15 51 1875 + 125 191 210 + 20 31 2125 + 125 136 255 + 25 27 2375 + 125 121 310 + 30 17 2625 + 125 82 375 + 35 1 2875 :±: 125 72 455 :±: 45

As l1as been pointed ou t theoretically/S( experi mente.lly/9/ and Ьу means of а computer aimulation/ 101, t he total range of &ho­\ver electr ons ; L; ni, i a proportional to the energy Е t of а gam­ma-quantum generaiing shower, e.g.,

Er =o(,2: R., /1/ ' ~ where d = (0 . 59 ± 0. 02) MeV/= f or liqui d xenon/9, 101. It is ahown also t hat wi th а good accuracy more general relation ia true:

..6 e( Ef' E~ = #'} • ..12:r .~~ , / 2/ ~ А Р t ~tAp

г---

3 1 j о•

1' •. ~ uс:~л.,,. .. 6:.../t:.' 1 ~ .

"- ~-----~

where f(Er,E0,t,p) ~ Ae(~f'~opt,p) are SEIL inside а square

~tAp. The coefficient ~ is conвtant to wi~hin about 3% and equal to the constant oG multiplied Ьу the average coefficient k of shower electron range projection enlargement (k =~Ri/ ~ri)/11 /

Тhе function f(Er,E0,t,p) may Ье defined in the form of а

product of conditional distributions:

r< Er, E0,t,p) = r 1 CEr,E0 ,t)·f2 (в1 , E0 ,p/t), /3/

where functions f 1 and f 2 describe а longitudinal (f1) and tranв­versal (f2 ) development of ES in а picture projection plane. The corresponding distributions in а space are related Ьу the analo­gous formula:

F(Et"E0 ,t,~) = F1 (Et"E0,tl•F2(Ef,E0 ,f / t), /4/

where ~ is а distance from а DSA and t has the same meaning as earlier.

Naturally enough to admit that

F 1 '(Е;, Е0 , t) ,. f 1 (Et, Е0 , t), /5/

and the normalization condition: -_,[F1 (ЕТ'Е0 , t)•dt .. Е f"' /6/

Тhen owing to axis symmetry (in average) of ESs developing inвide а uniform medium and te.king into account of the photographing conditions of ES in ХВСs we have/121:

1 О() df

f 2(Ef,E0 ,p/t) = 2• F2(E;,Eo•.f /t)• • о /1-(p/f )2

From /3/ - /7/ it follows that

Hence

2 [;;о~,...Е0 ,р/ t)•dp = 1, vt><>

2-fj F2 (EpE0 ,f/ t)fdf = 1. о

F(E1 ,E0 ,t,~) авс~,Е§.,t,~> 2 ~А tlt

171

/8/

/91

/10/

is an average value of SEIL inside а ring of the thickness ~~ along the radious ~ , measured from а DSA; i1t is the thickness of а ring along а DSA at the point t.

4

l t

), LONGITUDINAL DEVELOPМENT OF AN ES

Тhе distribution of SEIL along а. DSA has been appro:ximated Ьу the function:

r- 1 t 2 F1(Er,E0,t) = "'-t exp/-2("'f0) /, /11/

where the parameters с(. , r , and t 0 depend on Е Г and В0• It turned out to describe the dependence on Е Г Ьу the following simple functions:

У- -а2·Е~ 0 (ЕрЕ0 ) = а1 •е t + a 3•Er + а4 , /12/

toCEr,E0 ) = ь 1 ·lnEt - ь2• /13/

Because of the nprmalization condition /6/ we have:

ое - 2<1- r>J2.o.588-r. • - rc1 ~r>·t~+· r· /14/

Numerical values of conвtants а1 (i=1, • •• ,4) and ь1 (i=1,2 ) are presented in taЬle 2.

ТаЬlе 2 Numerical values of constants ai (i=1, ••• ,4) and ь1 (i=1,2) of the formulae /12/ and /13/, and corresponding values of probaЬility Р according to а 1(,-Z cri terion

i Е0. ~ 1. 5 МеУ Е0 = 3 МеУ

ai bi ai bi '

1 0.4 + 0.2 0.87 .± 0.04 5.6 + 11. 8 0.89 + 0.04 2 (5 : 4)•10-J 2.95 ± 0.16 (36: 26)·нrз з.о5: о.16 3 (29; 2)'10-5 (30; 1)•10-5 -

4 0.58 .± 0.05 0.5) ± о.оз

р = 0.46 р = 0.99 р = 0.45 р = 0.99

Note should Ье taken that at Е r ~ 500 МеУ the dependence of the parameter 't оп Е Г reaches practically i ts asymptotic beha-viour, i.e., .

i(Er,Eo) ';;: а)•Е; + а4. /12/

Тhе formula /11/ describes experimental data on the confiden­ce level of О . ). From /11/ it follows, in particular, that if а ES develops in an absorbent of the thickness t, then in average it is possiЬle to register the part E(t) of the energy E;of а gamma-

5

quaпtum geпerating ап ES. Thispa rt is :

E(t )

t

s F1 ( Е D

·1'/hеп the ratio

t 2k (1- 'f)f2 t oD k (~ )

, Е0 , ·r )d f = :~!; . 2 • (-) 1+~ __1:jJ_. • r ГС~) to k=O l"+1+ 2k 2Кk!

I(t ) = E (t ) / J!; r

/ 15/

/1 6 /

is sиificieпtly c1ose to uni t y СА ;:=: о . 6-О. 7), the ·tota1 eпergy

"' r of ап ES may Ьс eva1uated re1iaЬly eпough Ьу iteratioп method iпsertiпg iпto right -haпd part of /15/ а measured va1ue of eпergy

~meas ( t) iп р1асе of Е f as а first approximatioп.

4 . TRANSVERSAL ES D.i> VELGPI~lliNT

As а result of experimenta1 materia1 ana1ysis it has becn poiпted ou t t hat the Gaussian fuпctioп for SEIL transversa1 dis­tribution may Ье takeп at the energy iпterva1 Eff 700 LleV, i.e.,

f 2 (E r, E0 ,p/t) = N(O; ~2) . /17/

~t higher eпerGY va1ues some part of ioпization 1osses is co11ima­ted v1i th а 10\V dispersioп пеаr Ьу а DSA. The rest part has а signi­ficaпt1y gтeater di spertioп, о1 ~ 2. 5• о2 • Both parts may Ье approximated at the confideпce 1eve1 of 0.6 Ьу пorma1 fuпctioп so that a t Е 1 ~ 700 llieV 1'/е have:

f 2 (Er,Eo•P\t) ...---. 1

• (li(O;C)~) + a•N(O;()~)) . 21( ( (5'1 +аб'2 )

/1 8 /

Here а, 6'1 , апd 6 2 are the paro.meters depeпding оп t, J:;(• апd 1>0

апd beiп&determiпed Ьу experimeпta1 data statistical aпa1ysis. The dерепdепсе of these parameters оп t and Е r may Ье possiЬly

represeпted Ьу meaпs of simp1e fo1lowing functioпs:

Here ь 1

6', ~ 2

а =

m1•t + ь 1 ,

m2•t + ь 2 ,

-Б A•t , - 5 m1 = (0.05 .:t 0.02) + (3.6 .:t 1.1 ) •1 0 Er •

0.59 t 0.18, m2 = 0.04 t 0.01, ь2 = 0.27 .:t о . об,

6

/19/

j20/

/21/

/22/

.. 1 ..

(

,J

А = 39 .:t 24 , n = 1.42 .:t 0.1 0 , the energy Er i s expressed in MeV; the ·1ength t, iп rad. uni ts. Tl1e dep endeпc e of t l1e va1ues m1 and ь1 ( i = 1, 2) оп Е0 at experimenta 1 ac cura cy a chieved is not obser­

ved. Taking into account the trane.formatioп /71 v1e get a t t he

energy i n t er va1 Е 1 ~ 700 bleV the fo11owing expression for the pro­baЬi1ity distributioп fuпctioп F2 (Er, E0 ,t/~ ) of average SEIL iп а p1ane perpendi cu1ar t o а DSA iпtersectiпg this axis at а distaп­ce t from а primary gamma-quantum coпversion point:

F2 (Er, Eo•f/t) 1 -D2/2-2.

сб', ·е J !)1 а -r/ + 7"е / 26 2 \.) 2 2 ).

Correspoпdiпg relation at E t~ 700 MeV may Ье simp1y obtained from / 23/ v1heп · а = О.

5. GENERAL FORI.; OF SPATIAL DISTRIБUTION OF AVERA GE SEIL I N J::S

/ 23/

I t ari s e s from t he expressioпs /1 0 /, /4/, /11/, апd / 23/ tha t а distributioп of average SEIL re1eased

vo1ume ДV = 2 1Г~д~дt may Ье described a t

1vi thiп а ring of а

Е ;~ 700 JiieV usiпg the fo11ov1iпg fuпction:

2 2

L\E( Et, Е0 , t , ~ )

2.>7"fдfLit

2 1 -r1/2 а -r2/2 ,, 1 t" -112 -·е +-;:;r:•e "'1" е • IS"1 U2

2z ff't 0 ГCz ) а1 + а•б2 / 24/

Here z = (1 + f)/2; 1 = t/t0

, r 1 = f/ 6'1 , апd r 2 = 91 о2 are пorma1ized 1engths. The expressions for f, t

0, ~1 , and 62 ав

we11 o.s пumerica1 va1ues of corre~pondiпc constants are gi ven in sections 3 апd 4. At Е;~ 700 L;ev the fuпctioп /24/ simp1ifies consideraЬly because iп this case а = О. Тhеп the пumerico.1 va1u­es for the coefficieпts may Ье takeп from 151. Such а form as at а = О has the distributioп of average SEIL fo.r from shower maximum sinc e ап effect of а co1limatioп пеаr Ьу а DSA vanishes quickly.

6. LOIIGITUDINAL FLUCTUI.TIONS OF IONIZATION LOSSES

The esseпtia1 source of gamma-quanta eпergy measurement er­rors, wheп ioпizatioп effect of ESs produced Ьу these gammas is detected, are f1uctuatioпs iп 1ongitudina1 deve1opment of electro-

7

magnetic showers. То evaluate quantitatively such fluctuations it is suitaЫe to иве а distribution of the part A(t) of the sho­wer energy E(t) released inside an absorbent of the thickness t. Once assumed this thickness coincide with а cascade development length vle have:

A(t) = E(t)/Hf • /25/

Fluctuations due to а gamma-quanta conversion length distribution may Ье also taken into account as а proЫem of two known stochas­tic processes superposition. This proЫem is to Ье solved easily using а computer. The expresвion for an average value of A(t), ave­raged upon longitudinal SEIL fluctuations, is given Ьу the formu­lae /16/ and /15/.

It has been .ascertained . that the random variaЫe A(t) has approximately normal distribution, A(t)-N(A(t);()_i), at the inter­val A(t)~ 0. 5 which is of most interest from practical point of vie\v. The dependence of th~ relative dispersion . OA/A( t) . on A(t) is described Ьу the relation

1 а 1 ; 2

oA/A(t) = <ъ· ln --) , /26/ A(t)

which is valid within the limits АЕ/0.1; 0.95/ at the confidence level 0.3. Пеrе

а = oG - {)•Er , /27/ f1.. • -5 -1 and ~ = 1.02, 1• = 1.3 -10 MeV , Ь = 13. 6 ± 1.3.

Therefore the value E(t) is submitted to а normal distribution at A(t) ~ 0.5 also, во that

E(t) ~ N(!(t); (б'А:Е'(t)) 2 ), /28/

where E(t) is determined Ьу the expression /1·5/; ОА' Ьу the formulae /25/ and /16/.

Тhе relation /28/ describe.s in average the distributi on of the part E(t) of the sho\ver energy Е r released as SEIL along the absorbent thickness t (measured from а gamma-quantum conver sion point) within the shower energy E(t) is observed.

7. CONCLUSIONS

The results of investigation of shower electron ionization losses in electromagnetic cascades produced in liquid xenon Ьу

8

gamma-quanta with the ~nergy Er = 60- 3000 MeV may Ье sumшarized ав follows: 1. The distribution of average shower electron ionization losses

along а cascade development axi~ is described Ьу the formula /11/ together with /12/, /13/, and /14/. In particular if some part E(t) of shower energy is registered in an absorbent of the thic­kness t, then the primary gamma-quantum energy Er is to Ье eva­luated using the relation /15/. Such an estimation is the more reliaЫe the greater is the energy Er and the closer to Et is the measured energy part E(t), i.e.,the hlgher is the thickness t of an absorbent.

2. Transversal shower dimensions may Ье evaluated using the formu-A

lae /22/, /18/-/21/. At Er~ 700 MeV ionization losses collima-tion near Ьу а shower axis has been observed which diminiahes with increasing the distance t from а primary gamma-quantum con­version point.

3· Longitudinal ionization losses fluctuations which are commonly an essential source of errors, when an energy of gamma-quanta generating showers is to Ье determined, describes the normal distribution /28/. А dependence on the length t and the energy Er of а dispersion of this distribution is approximated Ьу the formulae /25/, /16/, and /15/.

The distributions quoted here may Ье used also in the case of others sufficiently dense absorbents as well as when not only

v ionization bu~ other effects are registered (for example,Cerenkov radiation). Finally it follows to note that so far as the experi­mental material used in this work has been obtained Ьу means of the detectors of dimensiona limited (the length of the xenon bub­le chamber of !ТЕР does not exceed about 25 rad. units) longitudi­nal shower dimension determined using our formulae may turn out to Ье а little shorter than the real one.

REFERENCES

1. Heitler w. Тhе Quantum Theory of Radiation (The Clarendon Press, Oxford, 1954).

2. Ivanenko I.P. Electromagnetic Oascade Processes. Edited Ьу Moscow Univ. Moscow, 1972 /Russian/.

9

3. Nelaon W.R. In: Computer Techniquea in Radia tion TranQport and Dosimetry. Edited Ьу W.R.Nelaon and T.M.Jenkina. Ettore Мajora­na Int ernational Science Seriea. Plenum Preas. New York and London, 1978.

4. stowinski в. In: Proceeding of the International Meeting on ProЪlems of blathematical Simul ation in Nuclear PhyaicsResearches J INR, D10-81- 622, Dubna, 1981.

5• Stowinski В., Czaj Vl. Izv, AN SSSR, ser,fiz. , 1981, v.45, N. 7 , р . 12)0.

6. stowinski в. at al. JINR, Р1-В2-689, Dubna, 1982; JINR, Р1-82-688, Dubna , 1982,

1. Niczyporuk в. at al. JINR, Р-2808 , Dubna, 1966. 8 . Rossi в. High-Energy Particle& (Prentice-Нall, Inc., New York,

1952). 9. Konovalova L.P. at al. JINR, Р-700, Dubna, 1961; Pri bory Techn.

Eksper. , v.6, 1961, р.261 /Rus sian/. 10. Borkovski :М . J , , Kruglov S.P. Report LIJP AN SSSR, N. 184, Lenin­

grad, 1975 . 11. Bancerek А. at al. JINR, Р1-81 -599, Dubna, 1981. 12. ::;towinski в., Czaj w. JINR, Р1 -80-341, Dubna, 1.980.

Received Ьу PuЪlishing Department on June 18 , 1984.

10

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EC81Ji)~;-~~-- ·UNep 1)'ИJ1И И И'1'ЭФ /)Aocda/.

_"_.ro нacтll'rY'I'a адаРJЬIХ иc~в.aaldt. Дубна t984

st~W!'!:~?~.t ' al. . -. EJ-84-418 fh~l~t ··~c~iption of the ~atial· Distribution of ~Ы.а~~ .. 1L()аеев ~n Electr011agnet~c Shovers Proc;luced Ьу . , .~ · : ' '.·~;.~itb ' the Ener8}' lyoi 6().3()90 К8V

tJur .!Ф•Р : .--~,~f aцalyeis of "eJ:Pe.timent a11y •asuied •hovп etectt;qa' !Oil~a~~op · losses in e~ectr0111aptic aЬovere p'r,'odueed Ъу ,-...-~- ,n.ta the eurgy __ _щ ~60-ЭСJОО ИеУ iц ltquid ~ •r• pr.._t .. ~~ <~ fo~1ae have t!eu.~ CJhtained detcriЫna ·t CJa-. git:Uclina1 ..US· -,:;·~raal avera..- lOD).&atiCJn lo•••• df-.tri'Ьu"' ti.Cщa ' ав wel1/!a~-f1.uctvations of eae:tgy . i~iaation loeaee ()f .!Щ~~~i-ona at.~: фе .Ь~r developмnt ахе8. n.e •.Wi.J~Мmtal materia.l ia oЬuhted usina pictu,es frOII xenon ЬuЬЬ1.• cЬ.-.rs of . .JIR and -IПР (Мoacqw).

-~- in~estigation hali been· prfoa.d at the J.abora.tort of eo.,uttai' Techniquea and .Aut:OII8tion •. JIRit ·4114 ~t t'Ье ·"-Yaical tuti~':'te of the Varsav Technical Uni~raity.

he;t'-11.~ 'of tbe Joint Inatittrte for Huct..ear leeeaJ"cb, DuЬna 1981t