A new simulational approach to electron-photon showers in heterogeneous media

Nuclear Instruments and Methods in Physics Research A257 (1987) 155-176

155North-Holland, Amsterdam

A NEW SIMULATIONAL APPROACH TO ELECTRON-PHOTON SHOWERSIN HETEROGENEOUS MEDIA

Mitsuhiro OKAMOTO and Toru SHIBATADepartment of Physics, Aoyama Gakuin University, Chitosedai, Setagaya-ku, Tokyo, Japan

Received 8 December 1986 and in revised form 13 February 1987

1. Introduction

0168-9002/87/$03 .50 © Elsevier Science Publishers B.V.(North-Holland Physics Publishing Division)

Three-dimensional Monte Carlo calculations are developed for determining the energies of electron-photon showers detected inemulsion chambers . The calculations provide better accuracy and higher precision by taking into account contributions from theLandau effect, the transition effect, etc., which are hard to solve analytically.

The simulation is applicable, even in the extremely high energy region ( -1000 TeV), for any type of chamber design, either withwide gap or alternate mixed substances . We present here two powerful ways to save computing time without sacrificing the accuracyof numerical results . The first employs approximate formulae for the cross sections of bremsstrahlung and pair creation processes,including both the screening effect and the Landau effect . The second uses analytical formula of multiple Coulomb scatteringapplicable when including various electromagnetic processes, such as bremsstrahlung, pair creation and ionization loss.

We compared our results with FNAL data and found remarkable agreement, both for average cascades and for fluctuation . Weconclude that the position of sensitive materials inserted between heavy elements critically affects the precision of the energydetermination.

Both accelerator and cosmic-ray experiments are requiring larger sizes for looking into deeper strata ofelementary particles . With this upward scaling, it is expected, particularly in the latter kind of experiments,that emulsion chambers (hereafter called ECs), which were first proposed by Kaplon [1] and laterdeveloped by Japanese physicists [2], will play an increasingly important role . Until now, various ECexperiments have been performed and brought fruitful contributions to the field of elementary particlephysics as well as that of high energy astrophysics, e.g ., for predictions of large transverse momentum [3],Centauro phenomena [4], X-particle [5], etc.The energy determination of high energy electrons (photons) by ECs is based on three-dimensional

cascade theory, which was developed first by Molière [6] and Bethe [7], and finally established byNishimura and Kamata [8]. Nishimura [9] performed further numerical calculations for EC projects, andmost EC groups are still making use of his analytical curves .

One should, however, note that the cross sections of various elementary processes used in the analyticalcalculations are always much simplified in order to avoid mathematical complexity. Some problems notconsidered in Nislümura's calculation may be summarized as follows:

(a) The Landau effect, particularly important for heavy materials in the region >_ 10 TeV.(b) Effects of ionization loss and deviation from the complete screening cross section, effective in the

region < 1 GeV.(c) The transition effect, mainly that arising from the inhomogeneity due to insertion of photosensitive

materials (X-ray films and nuclear emulsion plates), which is usually called the "spacing effect" .(d) The effect of "immediate gap", i.e ., the effect of the gap between the bottom surface of the

overlying dense materials and the setting position of the X-ray film or the nuclear emulsion plate inserted,where the number of electron tracks is actually counted.

(e) The effect of alternate mixed-substances, i.e ., the mixture of dense absorbers, such as carbon, leadand so on, makes cascade diffusion very complicated.

(f) Fluctuation problems.

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M. Okamoto, T. Shibata / New simulational approach to electron-photon showers

Although Nishimura solved approximately problem (c) in an analytical way be introducing the concept"spacing factor", a more exact solution is necessary for using ECs for wide gap-type .A further problem caused by the existence of gaps may bring breaks of scaling laws on primary energy

and lateral distance, which is essential for the energy determination in ECs . In fact, Hotta et al . [10] reporta small break of the scaling law even within the rather narrow variance 50-300 GeV.

Since it is hard to solve problems (a)-(f) in an analytical way, we perform the calculation by means ofMonte Carlo method .

Pioneer work of the Monte Carlo application to this field was done by Wilson [11], and later Messel andhis collaborator [12] developed it extensively . The energy region studied by them was, however, limited toat most several tens of GeV, because they were investigating the behaviour of low energy cascade showers,where usual cascade theory is not applicable .

Application of the Monte Carlo method to cascade showers in the TeV region was made by Adachi etal . [13] . The purpose of their work was to investigate the high energy photons and electrons in anatmospheric cascades shower (called "air family"), so that the threshold energies of individual secondaryparticles were on the order of 1 TeV . Another problem in their calculation was that the cross sections ofelementary processes were much simplified in order to reduce computing time .

Misaki et al . [14] developed the Monte Carlo calculation somewhat further by including the Landaueffect for various substances, such as lead and air . It is, however, not so easy to apply their method to apractical EC experiment, taking account of the problems mentioned in (b)-(e) .

Kasahara [15] performed extensive Monte Carlo calculations, taking account of the various effects(a)-(d). His calculations are mainly focussed to simple types of EC, because his purpose was to apply themfor the EC project at Mount Fuji .

In the accelerator community, on the other hand, a simulation program called EGS (SLAC-2110, 1978)is now widely used, and is the standard at accelerator laboratories all over the world . Its direct applicationfor cosmic-ray observation is, however, not so straightforward, since the experimental conditions inherentin the latter are somewhat different from those at accelerators.We present here three-dimensional Monte Carlo calculations that are applicable even in the extremely

high energy region - 1000 TeV. They apply to almost any type of chamber design either with wide gap oralternate mixed substances . They should, therefore, be particularly useful for intricate ECs, such as thosedesigned by the Soviet group and/or by the JACEE group.

In our calculations, the following elementary processes are considered : (1) bremsstrahlung, (2) paircreation, (3) ionization loss and (4) Compton loss .

For the lateral spread, we take into account only multiple scattering . One may worry about thecontribution from single scattering ; however, in practice it gives a negligible contribution to the spread ofthe electron shower observed in EC experiments . This is because the number of electron tracks is usuallycounted within radii of at most - 200 Itm from the shower axis, corresponding to 0.035 radiation length inlead, wherein the lateral spread due to single scattering is not effective in comparison with that due tomultiple scattering .We employ two powerful methods for saving computing time without sacrificing the accuracy of

numerical results. The first involves the construction of approximate formulae for the elementary processesof bremsstrahlung and pair creation including both the screening effect and the Landau effect . In sect. 2,we summarize the approximate formulae on these processes, and compare them with analytical onesobtained by Bethe-Heitler and Migdal. The second involves the construction of a formula for multipleCoulomb scattering that is applicable for including various electromagnetic processes such as ionizationloss, successive bremsstrahlung and pair creation . With the use of the formula, which is presented in sect.3, the three-dimensional calculation becomes quite efficient.

In sect. 4, we discuss the method of random sampling of various components, in particular on lateraldisplacement r and the deflection angle ,9 . Since the distributions of r and 0 are not mutuallyindependent, we must do their sampling carefully, taking into -account the correlations . To do so, atransformation to the principal axis is applied for r and 0 .

In sect . 5, we check the validity of the approximate formulae used in our simulations . We compare our

one-dimensional results with Messel's in the energy region 10 MeV to 50 GeV, and with Misaki's in theregion 100 GeV to 1000 TeV.

In sect . 6, we compare our three-dimensional results with FNAL data obtained by Hotta et al ., andthose of Sato and Sugimoto [16] .A discussion is given in sect . 7, particularly on the comparison of the present calculations with those by

Nishimura .

2. Elementary processes in cascade shower development

2.1 . Ionization loss and Compton scattering

The energy loss of electrons due to ionization during a passage of thickness At in units of radiationlength is approximately given by

where E is the critical energy of the medium, - 7 MeV in the case of lead. Throughout the present paper,unless otherwise specified, we use as the unit of length the radiation length Xo defined by [17]

with

-DE,/át= E,


157

Xo

137 A r° Z(Z+~)[In191Z-1/3 -f(Z)I,

(2a)

~=1n 1440Z-2/3/In 191Z-1/3 ,

(2b)where Z and A are the atomic number and atomic weight of the material concerned, respectively, N isAvogadro's number, ro the classical electron radius, and f(Z) comes from the correction term to the Bornapproximation, expressed by [17]

f(Z) = a2~

00

1

, a-~c2 .n=1 n(n2 +a2)

On the other hand, the energy loss of photons due to Compton scattering is given by [18]JEY _ _v 137

me

2Ey

51/3 -

In- - _l

(4)at

4 Z+~ In 191Z -

f(Z) 1

Me

61where me is the electron mass . For instance, when a photon with an energy of 10 MeV passes through alead plate, the energy loss is

-DEy/dt - 0 .58

MeV,

which is one order of magnitude smaller than that for the ionization loss of electrons .In the present work, we follow both electrons and photons until their energies become less than 1 MeV.

2.2. Bremsstrahlung and pair creation processes

If the energies of electrons and photons are larger, say more than 1 GeV, they are attenuated bybremsstrahlung and pair creation processes, respectively. Generally, the forms of the cross sections forthese processes are too complicated to perform random sampling on energy of secondary particles,particularly in the energy region > 10 TeV. For these cross sections, we construct approximate formulaethat reproduce fairly well the analytical solutions derived by Bethe-Heitler [19] and Migdal [20] . Bethe andHeitler derived analytically solutions for these processes including the screening effect . In the energy

158

M. Okamoto, T. Shibata / New simulat:onalapproach to electron-photon showers

region >_ 10 TeV, the so-called Landau effect becomes important, and Migdal calculated its cross sectionin a quantum mechanical treatment .

2.2.1 . BremsstrahlungLet us denote the probability function for bremsstrahlung as u(Ee , x), where x is the fractional energy

EY/E, of the radiated photon . In fig . 1, we illustrate the fractional energy loss in the case of lead

- 1_ A Ee = flxu(E., x) dx,Ee

dt

o

where we used the Bethe-Heitler formula for the energy region _< 100 GeV, and Migdal's formula forenergies > 100 GeV .

Fig . 2a presents the differential cross section multiplied by the fractional energy x for various cases ofelectron energy .

Generally, the explicit analytical form of u(Ee , x) is quite complicated, so we introduce the followingapproximate formula

u(Ee> x) dx = (1 -SB)r(v)

[ln(xo/x)) v-t d[ln(xo/x) ]

(

Fig . 1 . Fractional energy loss due to bremsstrahlung, and totalprobability of pair creation per radiation length in the case oflead . Solid curves correspond to those obtained fromBethe-Heitler and Migdal, while dashed ones correspond tothose from eqs . (6) (radiation loss) and (8) (pair creation) .

x = E#/E,

Y= Ee/Ej

Fig. 2 . Differential cross section multiplied by fractional energy x ( = EY/E,) for bremsstrahlung process (a), and for pair creationprocess (b) . Solid and dashed curves are the same as in fig . 1 .

(a)

- : Migdalpresent

op = f >v(E� y) dy .

QP-

_q~

(1 -SP).


159

1 .0-, Qp(E,)

0.5-

Migdal

------- : present formulaF.

Pb

1011

1d21d 310 14

1015

1e,1d2

013

0 -

1dsEe (eV) E, (eV)

Fig. 3 . Fractional energy loss due to bremsstrahlung (a) for three materials : carbon (C), iron (Fe) and lead (Pb), and total probabilityof pair creation (b) for each case . Solid curves are those calculated by Migdal, and dashed ones are from eqs. (5) and (7).

to be consistent with the analytical solution of both Bethe-Heitler and Migdal in the energy range 1 MeVto 1000 TeV. Here, F(P) is the gamma function . The parameters SB , P and xo depend on both the electronenergy Ee and on the kind of substance. Of course, 1 - SB corresponds to the fractional energy lossdefined by eq. (5) . Explicit forms - of SB , P and xo are summarized in appendix A.

In figs . 1 and 2a, we compare the analytical results with the curves expected from eq. (6) . One sees thatthey agree excellently well over the wide energy range.

In fig . 3a, we present the fractional energy loss for lead, iron and carbon, in the energy region 100 GeVto 1000 TeV. We show two curves in each case, one from Migdal's analytical solution (solid curve) and theother from the present formula (dashed curve) . One finds again that the latter well reproduce the formerfor any kind of substance .

2.2.2. Pair creationLet us denote the probability function of pair creation as v(EY , y), where y is the fractional energy

Ee/EY of the electron . In fig. 1, we show the total probability of pair creation for lead, as calculated byBethe-Heitler for the energy region -< 1 TeV, and by Migdal for > 1 TeV

In fig . 2b, we present the differential probability function v(EY , y) for various photon energies .Analogous to the case of the bremsstrahlung processes, the form of v(EY , y) is quite complicated, so we

assume the following approximate formula

v(EY , Y) dY = ~(1 - SP)

vo( Ey , Y),r +y(1 -Y)

dY'

(8)

(T) y(1 - Y)[y 2 + (1 - Y)2 + 2y(1 -y)]

for EY < 1 TeV

(9a)vo(E.Y , Y) =

go1/-o('r)

for Ey > 1 TeV,

(9b)

where go(T) and 2"o(T) are given by setting y=0 in eqs. (57) and (58), respectively . The parameters SPand T depend on both the energy of the parent photon and the kind of substance. Explicit forms of theseare summarized in appendix A. In the region EY -< 1 TeV, v(Ey , y) is identical with the formula found byBernstein [21] . From eqs . (7) and (8), we get

(10)

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M. Okamoto, T. Sh:bata / New simulational approach to electron-photon showers

In figs . 1 and 2b, we compare the analytical results with the curves expected from eq. (8). The presentformula reproduces fairly well the analytical curves over a wide energy range.

Fig. 3b presents the total probability of pair creation for the three substances shown in fig. 3a, asexpected from Migdal's formula and the expression for up in eq . (10) . One sees again they are insatisfactory agreement with each other.

3. Formalism of successive multiple Coulomb scattering

3.1 . Multiple Coulomb scattering taking account of the ionization loss effect

Suppose an electron with energy Eo traverses some substance of thickness t . If the ionization loss isneglected, the distribution function of r and $ at depth t, where r and 0 are the lateral displacement andthe deflection angle from the shower axis, respectively, is expressed as [22]

Eq . (11) is obtained by solving the following transport equation under the Fokker-Plank approximation

giving a similar functional form to eq . (11) . Explicit forms of d, b and c are summarized in appendix B.

3.2. Multiple Coulomb scattering including successive bremsstrahlung processes

A high energy electron loses almost its energy through the bremsstrahlung process. On the other hand,eq . (17) is available only for continuous small energy loss, such as due to the ionization process. In thissubsection, we present a formula available also for large energy loss due to the bremsstrahlung process.

[17]

(14)

where a( ,&') d,9' dt is the scattering probability in an infinitesimal thickness dt, given by

2a(0) dö=

137 me d~ , (15)In 183Z-1/3 -f(Z) Eó 104

where me is the electron mass and f(Z) is defined by eq . (3).Now, let us put the term for ionization loss into eq . (14) to get

0La~

where e is the critical energy . The solution of eq . (16) is easily obtained as (see appendix B)

4, (EO , t ; ^2r, $)= exp [ -îa(dr2 -2br-$+c~2 )], - =ac-b, (17)77 2

(EO, t ; r, $)=cwexp[ -w(ar 2 -2br 19 +c~2 )j, (11)

17

where

a=0S, b=~0St, c= _OSt2 , =nost2 > (12)w

with

2

OS = Ez t (Es : scattering constant). (13)Eo

M. Okamoto, T. Sh:bata / New simulational approach to electron-photon showers

161

e (E, . . e,+t )

------------------------

ri+i

t,

t i+1

Fig. 4. Illustration of successive bremsstrahlung process at t,A, --------------

and t,, 1.

Let us consider the case wherein an electron with initial energy Eo emits photons n - 1 times at (t� r,)with deflection angle ,4, (i = 1, 2, . . ., n - 1) in a material of thickness t . Taking account of the geometricalcondition as illustrated in fig. 4, the net deflection angle and the net lateral displacement of the electronsuffered during a passage of thickness t,+1 - t, (= 4,) are given by ,9,+1 - 0, and r,+1 - r, - a,+9�respectively. Thus, from eq. (17), the distribution function of r,+1 and 19,+1 at depth t,+1 is represented as

~(E, A, ; r,+1 - r -dt 19 >> $1 +1-I&J-

(18)

Therefore, the distribution function of r and ,9 of the electron after passing a material of thickness t iswritten down as

0(Eo, El, . . ., En-1, t0, tl, . . . . t,-, ; r, ó) dr dón-1

=f . . .f II O(E� a� r+l-r-A,t9� l~,+1- i~ ,) dr.�d#, +l,

where we put

{ to,r0 , ó0 } -{0,0,o},

{ tn,r� #n } = {t,r, 15 } '

In eq. (19), integrations with respect to rn (=r) and $n (=$) must, of course, be reserved .Eq . (19) is easily summarized in a compact form as presented in appendix C :

(19)

<D(n ; r, $)= w2exp [ -con(anrz-2bnr-ó+cn~Z) ]1

1=anCn -bn .

(20)Con

Here, for the sake of simplicity, we represent the variables Eo , E. . . . . . EE,_1, and to , t l , . . . , to- 1 by thesymbol n. Explicit forms of an , bn and cn are presented in appendix C.

3.3 . General formula for multiple Coulomb scattering including both bremsstrahlung and pair creationprocesses

Eq . (20) is still not convenient for practical applications in three-dimensional simulations, since bothpair creation and bremsstrahlung play essential roles in the cascade development. The modification neededfor the pair creation process is, however, straightforward with the use of the method presented in the lastsection .

Following Bhabha in Heider [23], let us call the incident electron the zeroth generation electron, and allphotons radiated directly from the zeroth generation electron first generation photons. Furthermore, wecall all pairs of electrons and positrons produced through the pair creation process of first generationphotons first generation electrons (hereafter we do not discriminate between electrons and positrons, andcall both of them electrons) . Similarly we call photons radiated directly from the (m - 1)th generationelectrons mth generation photons, and the electrons produced through the pair creation process of the mthgeneration photons mth generation electrons, etc.

162

M. Okamoto, T. Shibata J New simulational approach to electron -photon showers

Table 1"Root" of the mth generation electron before its birth with production vector {t(0, m), r,� , $,� )

Suppose that some electron in the electron "family" of the mth generation is produced at depth t(0, m)with lateral displacement rm and deflection angle tare by the pair creation process due to one of the mthgeneration photons. Let us call the geometrical combination {t(0, m), rm, ,4m } the production vector .Suppose further that the mth generation electron exits the rest of the material after radiating photonsnm- 1 times.

Now, without loss of generality, we can designate the "root" before the birth of the mth geneselectron concerned, as shown in table 1 . Of course, for the zeroth generation electron (= primary electron),we must put ( t(0, 0), ro , 00 } = (0, 0, 0} . The parameter n, (j = 0,L..., m) in table 1 is introduced, asillustrated in fig. 5, to specify the photon due to the njth bremsstrahlung emission among many photonsradiated from the jth generation electrons .

The net deflection angle and the net lateral displacement of the j th generation electron during thepassage from its birth-point t(0, j) to the n~th bremsstrahlung point t(n 1 , j) are given by

respectively, where

AP(i, j) = t(0, j + 1) - t(i, Í )>

áB(' , j) = t(l, j) - t(0, j)'

Q bremsstrah(ung point

X pair-creation point

- electron

WANIW photon

1- r - áp(n. , j)l,+1 -A B(nr , j)#,,

"t

t(0 .() W.)) t(2 . j ) Un,-(,j) t(n p i )

t(0 .1 .1)

-------------- ti8{n1 .) )---------------- -- _ap(n( .)) -- ->

tion process .

(21)

(22)

Pig. 5 . Illustration of successive brernsstrahlung and pair crea-

Generation Kind of particle Production vector

zeroth electron { t(0, 0), ro , 60 )Ist photon (t(no,0), ,(('),

1st electron { t(0, 1), r1 , X91 )

2nd photon { t(n i,1), r (2), 02 )

2nd electron (t(0, 2), r2, 02)3rd photon (t(n2, 2), r,(3),

1 th electron { t(0, j ),(j +1)th photon {t(n . , J), ry ),0"1)

(m-1)th electron {t(0, m-1), r.- 1, öm_1}mth photon {t(n._1, m -l), r,('), $m}

mth electron Wo,(0, n t), r., 0.)

So the distribution function of the electron corresponding to the "root" presented in table 1 is writtendown, with use of eqs . (20) and (21), as

*,(n o , n l , . . . . n�, ; r, ó) dr d$m

M. Okamoto, T. Shibata / New simulational approach to electron - photon showers

163

=I. . .

J11 0(n, ; r+1-r, -dp ( n., j)ó,+1 - dB ( n. , j)#,, &j+1 - ó,) dr+1 d#� 1 .

Here, we put {t(0, m + 1), r�,+1, O,�+1) _ ft, r, 0), and t(n �� m) --- t(0, m + 1) = t, i.e., Ap(n �, ; m) = 0,and AB(n �� m) = t - t(0, m) .

Similarly as in the case of eq. (19), eq. (23) is again expressed in the familiar form

e(m; r, ó)=

zexp~-Sl,� (A�,r 2 -2Bmr -$ +C�,t9 2 )~,

~m - A�,C�,-B,� .

(24)

Here, we represent the symbols n o , nl , . . ., n n by m, for the sake of simplicity . Explicit forms of Am, Bmand Cm are presented in appendix C .

In this way, once specifying a one-dimensional "root" of the pair creation and the bremsstrahlungprocesses before the relevant electron exits the material, we can get the three-dimensional distributionfunction for r and ,9 at arbitrary depth t . Namely, we neither need to subsequently perform thethree-dimensional samplings on r and t$ at every bremsstrahlung point nor to follow photons andelectrons in a three-dimensional way to the observation depth t . We only need to perform randomsamplings on r and ,9 once at depth t with the use of the one-dimensional quantities A,,,, B n and C,,,which makes the computing time quite efficient .

3.4. Distribution function for photons

Eq . (24) provides the general formula for the distribution function of some electron just at the placewhere it exits the material. The distribution function for the photon at the same plane is also derived in asimilar way as before.

Suppose that an electron radiates a photon at depth t', and the photon exits the material withoutproducing pair electrons, as illustrated in fig . 6 . Since the one-dimensional quantities A', B' and C'(hereafter we omit the suffix m for the sake of simplicity) up to the depth t' is stored on "DISK" in theprocess of the computer work, the distribution function of the electron on r' and 19' at t' is given by eq .(24).

As the photon spreads by geometrical conditions only, we have a relation (see fig . 6)r = r' + T',9',

and

$=$',

with

T' = t - t' .So, substituting r' and $' into eq . (24), we obtain the following distribution function for the photon atdepth t

9Pl,(r, $)= 72exp [ -Q'{A'(r-T'$)2-2B'(r-T',9) .$+C',~2}1

.-------------- T--___---__-__,

(23)

= 2 exp [ -2(Ar2 -2Br-i4+C,~2)l,

(2s)

Fig. 6 . Illustration of photon's penetration from material withthickness t, where t' is the birth-point of the photon con-cerned.

164

M. Okamoto, T. Shibata / New simulational approach to electron -photon showers

where

A=A', B=A'T'+B', C=A'T'2+2B'T'+ C,

with 1/s1=1/2'=AC-B 2 =A'C'-B'2.

4. Method of random sampling

4.1 . Random sampling on longitudinal component

4.4.1 . BremsstrahlungAs discussed in subsect. 2.2., the energy spectrum of photons is inversely proportional to energy, so low

energy photons are numerously radiated from the electron . We must cut off such low energy photons, orthe computing time is divergent. In the present calculations, we omit photons with fractional energysmaller than 10 -6 , which is small enough for neglecting the trivial break of energy conservation (eq. (5)) .

Then the mean free path XB of bremsstrahlung is given by

= (1-8B) T(v + 1)[ln(xo/n)] v,

Bwith n = 10-6 .

Therefore the random flight path to photon emission is expressed as

ATB = -AB In w .

Here and in what follows, w means a uniform random number in the interval [0, 1] .On the other hand, the energy of a photon radiated from an electron is determined by the following

relation

(26)

4.4.2 . Pair creationThe mean free path X P is equal to the inverse of the cross section (see eq. (10)), so the random flight

path to the pair creation point is expressed as

and we present P(w, T) for EY -< 1 TeV in appendix D.

ATP= -AP In w .

The energies of pair electrons produced are sampled as follows

w= flo(EY, y') dy'1fo

1oEv, y') dy'. (29)

v

Putting the energies of the above two as Eé and E.- we, , get

Eef = z [1 ± P(w, r)] EY , (30)

where in the case of EY >J TeV, P(w, -r) is simply expressed as

+4T +1P

(e+ 1 e--1 fw, -r

)__ __ (31)t-1 ~"+1' 1+4T -1

w=f lu(Ee , x') dx'/flu(Ee , x') dx', (27)

,,

which leads to

EY = Eexo exp [ -wl/" ln(xo/,1)] . (28)

4.2. Random sampling on transverse component

In the stage of random samplings on the longitudinal component up to some observation depth t, wehave obtained the parameters A, B and C, which are stored on "DISK" during the computer process.Thus we can determine the vectors r and 15 of electrons and photons at the observation depth t from eqs .(24) and (25).

The forms of the distribution functions (24) and (25) are not convenient for performing randomsamplings on r and 0 simultaneously, because of the mixed angular-lateral distribution. In order toeliminate such a difficulty, let us apply the following transformation from (r, 0) to (R, ®)

Finally, from eqs . (32) and (37), we can determine the lateral displacement r and the deflection angle ,9at arbitrary depth t .

5 . Check of simulation calculation


165

5.1 . Comparison with calculation by Messel et al.

Messel and his collaborators [12] performed extensive simulations of electron-photon showers in theenergy region < 50 GeV, taking account of the exact cross sections for various elementary processes . Inthe present paper, on the other hand, we carry out calculations with the use of approximate formulae . Thusit seems important to compare our calculations with those obtained by Messel et al ., in order to see thevalidity of our approximate formulae in the relatively low energy region . The statistics of our simulatedevents are one thousand in all cases .

Fig . 7a shows the average number of electrons obtained from both methods for electron- andphoton-initiated showers in air . The energies of the primary and secondary particle and those of the

(0r

) .( UÛz Ull1®l, Uz +UZ =1, (32)

so as to get the following canonical form

S2(Ar z - 2Br - O + C02) = R z/Rá +192/02 . (33)

To do so, we should set

U1 = 2(1+ßl , Uz = 2(1-

äl , Ro° (Y - ß), Bo = (Y+ß), (34)

where

a=A-C, ß= (A-C)z +4B z, y=A+B . (35)

Thus we have

(dR d0r, ,9 ) dr d$ = (

e_RZ/Ró ) X ~ e-eZ/®ó)

, (36)irR ó ir0

which is a convenient form for the random samplings on R and ®.With the use of the polar coordinates R(R, SW and O(O, q)® ), we get

R = R o -ln w =, OR 2arw, (37a)

O=O0V%ln w, $a =27rw . (37b)

166


(a) Air

0 2 4 6 8 10

0 2 4 6 8 10t (c . u .)

t(c .u.)

Fig . 7 . Average transition of electron number initiated by electron and photon in air (a) and lead (b) . Open and open-dotted circlesdenote the transition obtained from the present simulation for the cases of electron and photon primaries, respectively, while solid

and dashed curves are those obtained by Messel et al. for the same cases,

secondary electrons are attached to the curves . Fig. 7b shows the same results given in fig . 7a, but for 1GeV primary particles in lead . One finds that the present calculations are in agreement with those byMessel et al . within 5% . So the present empirical formulae for the Bethe-Heitler cross sections including

mc0

ud

óóz

102

E, =10 TeV

á+ lol-

E�/Em= 10 3

(b) Lead

Fig . 8 . Average transition of electron number initiated by anelectron with energies Ep = 10 and 100 TeV, where the mini-mum electron energy Em to be followed is set so that Eo/Em= 10 3 . Open circle and triangle indicate the results obtained byMisaki et al ., while black circle and square are results of

10 -1 i

C . u .

Ellsworth et al . The three curves are those obtained from the0

10

20

30

40

present calculation .

6.1 . Chamber structure

M. Okamoto, T. Shibata / New simulational approach to electron - photon showers

167

0.5 cm{

5.2 . Comparison with calculations by Misaki et al.

6. Comparison with FNAL data

0.25 c m

12 layers

0 .1 cm cl

0.25 cm

24 layers

0.25 cm

TYPE A

TYPE B

TYPE C

- : lead plate

Fuji nuclear emulsion plateSakura N-type X-ray film

Fig. 9. Design of ECs constructed by Hotta et al . (types A and B), and Sato and Sugimoto (type C. The plates of nuclear emulsionare composed of 50 pm emulsion coated on both sides of 800 Am thick acrylic bases.

the screening effect are suitable for investigating the cascade development in the lower energy region (< 10GeV).

Misaki et al . [14] performed cascade simulations with the use of the exact Migdal cross sections, whichare effective in the higher energy region (>_ TeV) and we compare our results with those. In fig . 8 wepresent the average number of electrons as a function of depth t, obtained by both Misaki et a] . and us forlead at three energies, Eo = 10, 100 and 1000 TeV with Eo/E~�i � = 10 3 . In fig . 8 we also present the resultsobtained by Ellsworth et al . [24] in which they apply approximate formulae for the troublesome functions0(s) and G(s), which appeared often in Migdal's formulae .

One finds that the results are all consistent with each other within statistical fluctuations . We concludethat our approximate formulae are suitable in the higher energy region (> TeV) as well as in the lowerenergy region presented in the last subsection .

Recently, Hotta et al. [10] and Sato and Sugimoto [16] exposed various types of EC in FNAL electronbeams of 50, 100 and 300 GeV. They investigated systematically the three-dimensional development ofcascades in lead. The structures of the ECs used by Hotta et al . are illustrated in figs . 9a and 9b, and thoseused by Sato and Sugimoto in fig . 9c . Hereafter we call the three types A, B and C respectively. Oneshould note that in type C, two sheets of X-ray film, with polyester base of 150 g,m thickness coated withnuclear emulsion of 25 Itm thickness on both sides, are always inserted after the 10th layer . The number ofelectrons in the nuclear plates is counted at two positions downstream from the 10th layer ; one at 400 Itmbelow the bottom surface of the overlying lead plate, and the other at the top of the next lead plate.

168


E 0 = 300 GeV

(d .f.=1 .18)

Thickness(cm Pb)

6.2 . Average number of electrons within a circular area

EO = 300 GeV

(d f. =136)

Thickness (CMPb)

25pmZr

50

100

Fig . 10. Average transition of electron number within radii of 25, 50 and 1001am in the case of types A (a) and B (b) . Black and opencircles denote experimental results, while solid and dashed curves give results calculated at the upper and lower sides of the emulsion

plate respectively .

In types A and B, the electrons are counted at two positions ; at the exit of the upper lead plate and justbefore going into the next lead plate . These two nuclear emulsions are coated on the upper and lower sidesof an acrylic base . As noted by the authors of ref. [10], only electron tracks that have deflection anglessmaller than 10 ° are counted, in order to reduce the number of background electrons .

As seen from fig. 9, the dilution factors, defined by dividing the geometrical thickness of the EC(including both dense and photosensitive materials, and sometimes the air gap too) by the thickness of thedense material (lead here) alone, are given by 1 .18, 1.36 and 1 .48 for types A, B and C, respectively .

In figs . 10a and 10b, we illustrate the average number of electrons within radii of 25, 50 and 100 ltm forthe case of types A and B, respectively, when E0 = 300 GeV . One should note that the electron numbers atthe lower side of the emulsion plate (white circle) are 30-40% less than those at the upper side (blackcircle) . This is because the electrons have considerable deflection angles when exiting the lead plate . In fig.10, we draw two curves obtained from the present calculations ; one counted at upper side of the emulsionplates (solid curves) and the other at the lower side (dashed ones) . The statistics of the simulations are onethousand in all cases . In the present calculations only electron tracks with deflection angles smaller than10' are counted in order to be consistent with the experimental conditions .

All cases show that our calculations are in acceptable agreement with the experimental data, both inshape and in absolute value . It is noteworthy that the difference between the number of electrons at theupper and at the lower side are excellently reproduced by the present calculations . This result has neverbeen obtained quantitatively from the analytical calculations.

Fig . 11 presents the average number of electrons measured at the upper side of the emulsion plate forEo = 100 GeV . Again there is satisfactory agreement with the present calculations .

10 -?

10 -4

10 -5

M. Okamoto, T. Shibata / New simulatlonal approach to electron -photon showers

Eo =100 GeV

Ez\c 10_ ": .O ~Ó-

y

w _

a

d.f. = 1 .18

rEo/100 GeV (Nm)

Fig . 11 . Same as fig . 10, but for Eo = 100 GeV, measured at theupper side of the emulsion plate . Solid and dashed curves arethose obtained from the present calculation for types A and B

respectively .

z

10-3j

4 cm Pb1

T~~ \

0 -T 100

200 T3Ó0

Fig . 12 . Average transition curves of electron number within 50gm for energies Eo = 50 GeV (circle), 100 GeV (square) and300 GeV (triangle), obtained by Sato and Sugimoto, which aremeasured at the upper side of the emulsion plate. Solid curves

are obtained from the present calculation .

E f\ C,Cc 10"3

Mu

_ _z 10-3j ;"

-

mc0

óz

100-

E 50 .0

uwm

ó 5

10 -1

1 0-4

Thickness (cm Pb)

d . f. = 1 .36

b

10-1-0 100 200 300

r E,/100 GeV (Nm)

3 CMPb

169

Fig. 13 . Lateral distributions normalized to 100 GeV for types A (a) and B (b) at three depths : t = 2, 3, 4 cm Pb. Black and whitecircles present experimental data. The former correspond to 100 GeV and the latter to 300 GeV. The solid and dotted curves are

obtained from the present calculations, the former denoting 100 GeV, the latter 300 GeV .

170

M. Okamoto, T. Shibata / Newsimulational approach to electron -photon showers

100

50

100-

501

100J

50 I1

6.3. Lateral distribution

6.4. Fluctuation problem

O : r :s 50 N0 :

100

1d .f.=1 .36

o : r<<-50 N100

1d. f.=1 .18

ß/(n) (9'0)

a

-

- i _a- f' 100 GeV

i}*:~* 100 GeV

100 ~,

50 \

300

." .

Fig. 14. Dispersion of the number of electrons divided by the20 .

average number of electrons at each depth, for type A (white10

and black triangles) and type B (white and black circles) . Solid0

1

2

3

4

5

6

and dashed curves are obtained from the present calculations,Depth in lead (Cm)

corresponding to chamber designs A and B, respectively.

In fig. 12 we exhibit the results obtained by Sato and Sugimoto for energies of 50, 100 and 300 GeV,together with curves expected from the present calculations, where the number of electrons is determinedwithin a 50 ,um radius at the upper side of the emulsion plate. Again there is agreement between themeasurements and calculations.

In figs . 13a and 13b we illustrate the lateral distributions normalized to 100 GeV in the case of types Aand B, respectively, all measured at the upper side of the emulsion plates . The black and white circlesrepresent the experiment data, while the solid and dotted curves represent the calculations .

As pointed out by the authors of ref. [101, the scaling law does not hold at deeper layers, even at smalldistances from the shower axis . The present calculations satisfactorily reproduce the experimental points,although small deviations are found at a shallow depth, t = 2 cm Pb in the case of d.f. = 1 .18.

Hotta et al . investigated the dispersion of electron number at every layer, as shown in fig. 14 with theresults of our calculations, where the vertical axis denotes the dispersion divided by the average number ofelectrons. Minimum dispersion occurs near the depth of the shower maximum (see also fig. 10), asreasonably expected . As emphasized by the authors of ref. [10], the dispersion is far larger than thatexpected from a Poisson distribution at any depth. The present simulation results agree satisfactorily withthe experimental fluctuation, indicating that the random samplings on both interaction points and thesecondary energies are done correctly.

7. Discussion


Electron-pair primary

Thickness of Pb (C .U .)

0 2 4 6 8 10 12 14 16 18 20

Fig. 15 . Transition of electron number within a 50 pm radiusfor three energies, 1,10 and 100 TeV, in the case of electron-pairprimaries . The solid, dotted and broken curves are obtained bythe present calculations, each corresponding to counting posi-tions 0, 500 and 1000 tam from the bottom surface of theoverlying lead plate . The heavy solid curves were obtained by

Nishimura.

100

10

Fig . 16 . Transition of electron number within a 50 Wm radiusfor three energies, 0.2, 0 .5 and 1 TeV, in the case of electron-pairprimaries. The counting position is fixed to the bottom surfaceof the overlying plate. Solid and dotted curves correspond totwo cases, one regarding to the nuclear emulsion plate as an air

gap, the other taking it correctly into account .

Now we discuss the numerical results obtained by Nishimura and compare them with our calculations .We consider the chamber structure that Nishimura assumed in ref. [9], i.e., two sheets of N-type X-ray filmand one plate of nuclear emulsion are inserted alternately between 1 cm thick lead plates, corresponding tod.f. = 1.255 * .

In fig . 15 we show the transition curves for the number of electrons within a 50 ftm radius for variousenergies of electron-pair primaries. The solid, broken and dotted curves denote the results obtained fromthe present calculations, each corresponding to counting positions S = 0, 500 and 1000 ,um, respectively. Itis seen that the transition curves are very much affected by the counting positions. Also shown are theanalytical curves (heavy solid ones) of Nishimura. One sees a clear discrepancy in the energy region >_ 10TeV, which is attributed to the Landau effect . This discrepancy should here not be considered very severe,since the horizontal axis can be moved freely in practical applications.

Our calculations seem to be roughly in agreement with Nishimura's near the shower maximum in thecase of S = 500 Am. It should, however, be kept in mind that since the counting position of the electrons,either in X-ray film or in emulsion plate, is not definite in Nishimura's calculation, there is someuncertainty in a strict comparison of his curves with our results.

* The dilution factor 1 .255 is a little different from the spacing factor defined by Nishimura. According to ref. [91, the spacing factorcorresponding to the chamber design concerned is 1 .45 .

172


Our calculations indicate that the position of sensitive materials critically affects the precision of energydeterminations . Even the packing condition of photosensitive materials is essential, i .e . the setting order ofthose materials as well as the thickness of the black paper envelopes become serious * .

The present method enables us to perform simulation calculations of electron-photon showers within avery short computing time even for heterogeneous media, either with wide air gap or without any space. Inthe case of the chamber structure described in the last section, we can regard the nuclear emulsion platesimply as an air gap, since the thickness of each lead plate is much heavier than that of each sensitivematerial (- 0.0056 c.u .) . On the other hand, in the case of those consisting alternately of thin lead plates,for instance 2 mm (- 0.4 c.u .), and thick sensitive material, say 500 ltm nuclear emulsion (- 0.016 c.u .),the heterogeneous effect becomes significant . In fig . 16, we show transition curves of electron numberwithin a 50 ltm radius in two cases, one regarding the sensitive material merely as an air gap and the othertaking it correctly into account . One sees that the deviation is remarkable for deeper layers .

Application of the present simulation program (first version package) for an emulsion chamber ofintricate structure is performed by several authors [26], and more studies on the energy determination ofcascade showers, particularly induced by cosmic-ray heavy primaries, will be reported in the near future .

Acknowledgements

The authors would like to express their sincere thanks to K . Yokoi and I . Ohta for valuable discussionsand comments . The acknowledgement is also made to W.V . Jones for a careful reading of the manuscriptand valuable comments . We would also like to thank T . Yuda and N. Hotta, and Y. Sato and H. Sugimotofor kindly offering us original FNAL data on cascade showers, with valuable and helpful remarks .Appreciation is also extended to H . Semba, T . Tabuki and S . Kamata for their helpful comments onelectronic-computer work . The computer used were the ACOS 950 (Aoyama Gakuin Univ.) and theFACOM M-180 II AD (Inst . for Nucl. Study, Univ. of Tokyo) .

Appendix A

Summary of explicit parametrization for approximate formulae

Here we summarize the explicit parametrization of coefficients appearing in the text . In the following,the radiation length Xo is measured in centimeters, and the energy in TeV.

In table 2 we show the explicit forms of v, Xo and T, where we put

T1e = U/Ee,

11y = UIEy ,

~e = Ee/Xo,

~y = E,e1X0«

(38 )U is a parameter introduced by Bernstein [21], depending on the atomic number Z, which is given by

U=mecz

255

(39)Z113 15 .6 - ~ln Z

In table 3 we give the forms of SB and SP , where go(11y ) = g(0, rl y ) (see eq. (57)), and we introduced afunction

fo(0=

1

3~[(1-n)z+6

n(1-

n )+19z][1+i-n 1 + -inInv } .

The variables EB,P, wB,P, etc., appearing in table 3, depend on Xo only, and are summarized in table 4.

* A similar consideration was pointed out by the group of Louisiana State University [251.

Table 4Summary of variables appearing in tables 2 and 3

Appendix B

Solution of the transport equation

ô

ó

8

1 Es 2(at - p' 4 - EaE,)F = -4E2 q F,

where we defined

F(EO, t; p, q) =4 1r2

exp[ - á(&q 2 + 2bp q+cp 2 )1,

SB(Ee)

SP(EY)1=0

Applying the Fourier transformation for eq . (16) from r and $ to p and q respectively, we get

173

(41)

F(EO, t; p, q) = 41r2 J J e'(,.p+a'9 )

«Eo, t ; r, 9) dr dl9,

(42)

and we used the Fokker-Planck approximation on the integration with respect to 0' in the integrand ofeq. (16) .

Eq . (41) is easily solved as follows,

(43)


Table 2Explicit forms of v, Xo and T

Ee Y 51 TeV Ee Y > 1 TeV

v 1+0.657 190377 1-0.07 4 eXo 1-0.174,4 300 1T 19, 1 .5/X

Table 3Explicit forms of SB and SP

Ee,Y 5 E B,P EB,P < Ee,Y < EB,P EB,P < Ee,Yz

1-fo( 1)e) wB [ln(E./EB)] e alfln(Ee/EB)1`1=02

1 - igo(TI Y ) wP[ln(E.,/EP) ] " Y_ bllln(EY/EP)l1

Bremsstrahlung

'E B 0,10EB = 38 .5 X0O-444

Pair creation

EP = 10.0EP = 212 Xo 610

K = 2.700 Xó.250 =1.822 Xó.427

wB =1.950exp(-6.503 Xó'25) wP=1 .165exp(-3 .210 Xó45)

a0 =-0.06611n Xo +0.3630 bo =-0.640 Xó 108 +1al = -0.01481n Xo +0.1585 b1 = 0.199 Xó.045

a2 = -0.0260 exP(- 0.402 Xo) +0.07626 b2 =169

-0.0138 X00-

174

M. Okamoto, T. Shibata / New simulateonal approach to electron-photon showers

where

â = ©s ,

with

©S = 1 + S (E)2t,

(44a)

b=8L(1+8)ln(1+8) -11ét,

1

2~(1+ 1)i

]ó2 2ln(1 + 8) - 1

5 t-91 _é

},

8=-t and E=Eo -et .

From eq. (44a), one finds

s©S (t) =

Er', t,EO E(t)

Here F in the above equation is given by eq. (43) after replacing Eo , t and q by E� 4� q +respectively.

The right hand side of eq. (49) is easily summarized, with the use of eq . (44), as

wheren-1 n-1

n-11: 2

an

E ar, bn- E (a,T+1+b,), Cn-

(a,T+1+2b,T+1+C,),i=o i=o

i=o

(44b)

(44c)

(45)

namely, comparing the above with eq . (13) in the text, the mean square deflection angle in the case ofincluding ionization loss is given by the replacement of Eó by EOE(t).

Applying the inverse Fourier transformation for F from p andq to r and $, we obtain eq . (17) .

T+1P,

JO (n ; P,4)=

14~r2exp [ -á(ang2 +2bnp - q+c � p 2 )

j,

(50)

(51)

Appendix C

Derivation of eq. (20)

Let us apply the Fourier transformation to eq . (19) from r and ,9 to p andq respectively .

JO(n ; P, q) = 41r2 e'(r.p+o-q)O(n ; r $) dr d,i. (46)JJ

With the following modificationn-1 n-1

r'P+# - q= (r+1-r-di$j .P+ 1: (10, +1-#,)'(q+T+1P), (47)1=o i =o

where

and T,=t-t� (48)

we getn-1

J.(n ; P, q) = (47r2 ) n-1 fl F(E, A, ; P, q+ T� ,p) . (49)=o


and d� b� c, are given by putting the suffix i to the variables in eqs. (44a)-(44c),

175

(Ô, t, 8, EO ) - (Ó� 4 � S� Et) .

(52)

Applying the inverse Fourier transformation to eq . (51) from p and q and r and 0, respectively, weobtain eq. (20) .

In a similar way, we can get eq. (24), where we put

and further, corresponding to the replacement in eq. (52), we must change the suffic i into i, j in allvariables appearing there.

= L1+3Yo

(Hó +8) -Ho -3

(Hó +8) +HOJ

,

y=-12 [1+P(YO , w, T)] >3

P(y, w,T)=

(H l + 8) -H-3(H Z +8) +H .

The parameter T being extremely small except in the energy region Ey -- U (= 3 MeV for lead), we cansolve eq. (55) by the iteration method, i.e ., putting T = 0, the solution is immediately written down as

(59)

with Ho = H(y, w, 0) = -Z(2w - 1) .Since T is very small, H(y, w, T) does not depend strongly on y, so that putting y =yo in H(y, w, -r),

we get(60)

(61)

Putting yo =y iteratively in eq . (60), we finally obtain the solution of eq. (55). Practically, only a fewiterations are enough for Ey = 10 MeV to 1 TeV. In the region 1-10 MeV, the solution is tabulatednumerically.

Finally putting P(w, T) = P(yo, w, T), we can get the electron energies Eel as shown in eq . (30).

Appendix D

Explicit form of P(w, T) for E,, <_ 1 TeV

Eq. (29) leads to following transcendental equation

4Y3 -6y2 +9y--f+H(y, w, T)=0, (55)

where we introduced the following functions

H(y, w, T)='2+9[g(o, T)w - g(Y, T)J, (56)

g(Y, T) 7 4=9 + 3T [1 -Y - (T + á4(y, T)] , (57)

In ~-1+2y ~-1,

~= 1+4T . (58)

� , m mE a(nJ), Bn= L b(nJ ), Cm = L c (n,), (53)J =o J=0 J=0

and we perform the following replacements in eq . (51),

(an , bn , cn) -, (a(nj) , b(n,), c ( nj)) , (54a)

(ar , br, ci) - kJ9 c"J) , (54b)

176

M. Okamoto, T. Shibata / New simulational approach to electron- photon showers

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Japan-Brasil Emulsion Collaboration, Can . J . Phys . 46 (1968) 660 ;T. Shibata, Phys. Rev. D22 (1980) .

[4] C.M.G . Lattes, Y. Fujimoto and S . Hasegawa, Phys . Rep. 65 (1980) 151 .[5] K . Niu, E. Mikumo and S . Maeda, Prog . Theor . Phys. 46 (1971) 1644.[6] G . Molière, Z. Naturforsch . 3a (1948) 78 .[7] H.A . Bethe, Phys. Rev. 89 (1953) 1256 .[8] K. Kamata and J . Nishimura, Prog. Theor . Phys . Suppl. 6 (1958) 93 .[9] J . Nishimura, Prog . Theor . Phys . Suppl . 32 (1964) 72 .

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J .C. Bucher and H. Messel, Nucl. Phys. 20 (1960) 5 ;D.F . Crawford and H . Messel, Phys. Rev. 128 (1962) 2352 ;H . Messel, A.D. Smirnov, A.A . Varfolmeev and D.F . Crawford, Nucl. Phys . 39 (1962) 1 .

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J . Nishimura, Handbuch der Physik, ed., S. Flugger (Springer, Berlin, 1967) vol. 46/2, p.l .[18] W . Heitier, The Quantum Theory of Radiation (Oxford, 1956) p . 211 .[19] H.A. Bethe and W. Heitier, Proc . Roy. Soc. (London) A146 (1934) 83 .[20] A.B. Migdal, Phys . Rev. 103 (1956) 1811 .[21] I .B. Bernstein, Phys . Rev . 80 (1950) 995 .[22] L. Eyges, Phys. Rev . 74 (1948) 1534 .[23] H.J . Bhabha and W. Heitier, Proc . R . Soc. (London) A159 (1937) 432 .[24] F.W . Ellsworth, R.E. Streitmatter and T . Bowen, Proc. Int. Conf. on Cosmic rays, Kyoto (1979) vol . 7, p . 55 .[25] L.G . Porter, L.B . Levit, W.V . Jones and S.C. Barroes, Nucl . Instr . and Meth. 126 (1975) 69.[26] Y . Funayama and M . Tamada, J. Phys . Soc . Japan 9 (1986) 2977 ;

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A new simulational approach to electron-photon showers in heterogeneous media