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Analytic Treatment of Proton Induced Electromagnetic Cascades.

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Annalen der Physik. 7. Folgr, Band 33, Heft 5, 1976, S. 359-373
J. -4. Barth, Leipzig
Analytic Treatment of Proton Induced Electromagnetic
Cascades
By S. BARMO
l)
Institut fur Theoretischc Kernphysik der Universitat Karlsruhe
A b s t r a c t . The electromagnetic cascade induced by the no’sproduced by the interaction of a proton and its secondaries (mainly z*’s) is described.
Simple considerations indicate that a large part of the energy of the proton is used to producr
no’s (see Appendix B). Using the scaling hypothesis of FEYXMAU
we derive analytical solutions for
the cascade equations.
The solution indicate that a t the beginning of the cascade the no’s produced by the nucleon are
niorc important for the development of the cascade than those produced by x&’s. The situation ehangcs with increasing atmospheric depth. The average energy of the electrons and photons is nearly
the same as indicated by the above mentioned simple considerations. A comparison with a photon
induced casca.deshows t,hat the proton induced electromagnetic cascade is developing slowly and living
longcr.
Analytische Behandlung der protoninduzierten elektromagnetisehen Haskaden
I n ha1 t s u b e r s i c h t . Bei den Wechselwirkungen einrs in die Atmosphare einfallenden Protons
und der Srkundarteilchen mit den Atomkernen der Lufthulle werden zo produziert, welche elektroniagnetische Kaskaden induzieren, die sich uberlagern. Eine einfache Abschatzung zeigt, daB ein
groBer Teil der Energie des Protons fur die Produktion von no verbraucht wird.
Untcr Anwendung der Scaling Hypothese von Feynman werden analytisrhe Losungen fur die
Diffusionsgleiehungen hergeleitet.
Beim Vergleich mit eirier photoninduzierten Kaskade zeigt sich, daB sich die protoninduzierte
Kaskadr laugsanier entwickelt und zerfallt. Die Losungen ergeben fiir die mittlere Energie der Elektronen und Photonen etw-a denselben Wert wie die einfache Abschatzung.
Am Anfang der Kaskade werden mehr no durch Nuklconcn als durch x i produziert. SIit zunehmcnder atmospharischer Tiefe wird das Vcrhaltnis n5 -Beitrag zum Nuklconen-Beitrag immer groBer.
1. Introduction
The primary cosmic rays are mainly coniposed of protons (S5O0) [l]. The protons
lose their energy in the earth’s atmosphere through interactions with oxygen and nitrogen nuclei. If a proton interacts with a nucleus x i ’ s and no’s are produced and the surviving nucleon will retain about 500,{Jof the energy. It can interact again and produce
more x*’s and xo’s.
I f the energy of the produced n*’s is high ( E , 2 Bd m 113 GeV) they can interact.
and produce secondary n*’s and no’s, otherwise they decay.
The produced no’s decay immediately (trio N 10-l6 sec) into photons, which induce
elect,romagnetic cascades.
l) Present Adress: University of Damaskus, Facult,y for Electrical and SIechanical Engineering, Damaskus, Syria.
I n the following, we shall be concerned with the overlap of these electroiiiagnetic
cascades.
To describe this process a coupling of the cascade equations for the hadronic cascade
to those of the electromagnetic cascade is necessary. The cascade equations of photon
induced electromagnetic cascade can be solved analytically in the so-called Approxiniation 9 and B [2, 31. For the treatment of the hadronic cascade assumptions about
the production of particles by strong interaction are necessary. MOIJERF, and BUDINI
[4] assuined a scaling law for the nucleons similar to the scaling laws of pair production
( y --f e+e-) and breinsstrahlung. We shall use the scaling hypothesis [ 5 , 61 to derive analytical solutions for the electromagnetic cascade induced by a proton. The simple considerations in Appendix B give an indication as to the importance of the electromagnetic
part of an Extensive Air Shower. We shall calculate the average energy of the photons
and electrons and compare the solutiomwith those of a photon induced cascade with
the same energy.
The effects of the two different ways of no’s production, namely by n+’s and by
nucleons, and the first order 715’s decay correction will be discussed.
2. The Cascade Equations
The coupling of the cascade equations of a hadronic cascade [7, 81 with those of a
photon induced electromagnetic cascade gives the following equations :
(2.la)
(2.1b)
m;
Bd
=
R .T
TOM
*
g
= f13.3GeV
for T
nO(E, t ) = 2-J Qnxo(E‘,E ) n(E,t ) dE’
A, E
=
216.2K [ I t ] ,
+ - / wnn0(E’,E ) z ( E ,t ) dE’ ,
1
-
Ldn
00
-- t , - f mpy(E’,E ) p ( E ’ , t ) dE’ - py . y ( E ,t )
at
E
n ( E , t ) is the average number of nucleons at the atmospheric depth t with energy
( E ,E
dE).
n,Y,P are the corresponding quantities for n+’s photons and electrons. nO(E,t ) is
dt). A, andA, are the attenuation lengths
the averagenumber of no’s produced in (t, t
for nuclens and pions measured in radiation lengths.
wij(E’,E ) is the probability that a particle of type i and energy E’ will produce a
particle of type j and energy E.
The known scaling laws [9] for wrad(E’,E ) and wpair(E‘,E ) (the probabilities
ceeded in Approxiination A) permit an analytical treatment of a photon or a n electron
+
+
361
Bnalytic Treatment of Proton Induced Electromagnetic Cascades
induced electromagnetic cascade. For the probabilities of the hadronic cascade we use
the FEYNMAN
hypothesis that for a reaction
a
+ B+ c + X
a30
for high energies (E‘ 3 CQ),E - is only a function of x, where x = Pgbf./Pc.M..
d3P
I n the energy region for which the solutions will be derived we have x % E/E’.
With this hypothesis and with w n o y ( EE, ) = 2 * @(El- E)/E’ [ l o ] the equations
(2.la)-(2.le) can be cast into the following form
1
--_
-g n n ( E / E ) n(EI,t ) - -n ( E ,t ) ,
An
at
03
8n(E,t)
-=at
1
g,,(E/E’) n(E’,t )
4E
- (1/&
+ B,/h’t
r%
+!An E’
cos 8 ) n(E,t ) ,
(2.2a)
g,,(E/E’) n(E’,t )
(2.2b)
(2.2c)
/i
(2.2d)
03
+2
---=
”(”
at
t,
g 8 ( 1 - E/E’
s [p(E,
1
-
t)
- 1/(1-
0
(2.2e)
3. SoIutions and Results
3.1. a0Induced Electromagnetic Cascade
I n this case, the equations (2.2) reduce to
(3.la)
(3.lb)
I n (3.la) we assumed that the no will decay a t t = to. We solve the equations (3.la)
and (3.1b) in a similar way to that used for the photon induced electromagnetic cascade
with the help of integral transforins, namely the Mellin and the Laplace transforms.
S. BARMO
362
TVe obtain
;Irf ( s ) , J4< ( s ) , iW:(s), M $ ( s ) , & ( s ) and & ( s ) are the sanie functions that arise from the
electroiiiagnetic cascade induced by a photon (see Appendix A).
The solutions (2.3a) and (2.3b) differ only by the factor 2(sT 1)-1from the solutions
for a photon induced cascade. They depend as expected only on (t - to).
m
In Fig. 1 r ( E ,A',, t ) =
[ y(A",En,t)dE' is shown as function o f t for different values
E
of the ratio E/Ez.
3.2. Proton Induced Electromagnetic Cascade with n* Decay Xeglected
In this case the equations for the hadronic cascade are given by
(3.3a)
-Applying the Mellin transforms to the eqs. (3.3) we get
%!I
(S>t)
at
=
- an(s)
*
fkf(87
t),
(3.4a)
(3.4b)
(3.4ca)
363
.4calytic Treatment of Proton Induced Electromagnetic Cascades
m
For @Jl(s,t ) =
W
jES- p ( E , t ) dE and ynl(s, t ) = .f
E8 y ( E ,t ) dE we get
0
0
(3.5a)
(3.5b)
The equations (3.4) and (3.5) can be solved using the Laplace transform. We obtain
+ M&(s) e - " m + j l p4P (8) e--a,(s)t].
~ qJ T ~
~~ ~ ,g Lwip,
~ Au~p,
, M ; ~ ~i~
,
and ~f~
yZxO(s),M,(s) and a,(s)
(3.6b)
m),Po,I&),
depend on A M ,
US),
gnno(s),
W
(see Appendix A). I n Figs. 2 and 3 F ( E , E,, t ) =
y ( E , t ) dE'
El
W
and P ( E ;Eo, t ) =
1p ( E ' ,t ) dE' are shown as function o f t for different values of E,/E.
E
They are divided into two parts, one part comes from nucleons and the other from
Z*'S.
3.3. Solutions with n* Decay Not Neglected
3.3.1. So1ut)ions
We solve the equations (2.2e)-(2.2e) wit,h the ansatz
(3.7a)
(3.7b)
(3.7c)
(3.7d)
(3.7e)
S. BARMO
364
(3.8e)
+ n)
5:(~,t ) = i n n o ( $
a?&,
-at
a@&,
-at
t)
- -Po
*
( n = 1,2, 3, ...),
%n(s,t ) * A,'
2
*
t , - -A($
Yn(&
4 + C(S + n ) Pn(& t ) + s + n +
+ n ).
a
&(S,
t)
+ B ( s + n)
5: (8, t )
1
(n = 0, 1, 2, ...),
*
Fn(S,
(3.8g)
f)
(n = 0, 1, 2, ...).
The initial values for the equations (3.8) are given by %(s, 0) = E8,, %,(s,
lias-
(3.8f)
0) = %:(a,
(3.8h)
0) =
Fn(s, 0) =f n ( 8 , 0) = 0. Of course yo(E,t ) = - ES+~yo(s, t ) is identical with y ( E , E,, t )
2nz.
of (3.6a), the same is true for po(E,t ) and p ( E , E,, t ) of (3.6b). Therefore we calculate
onlv
y,(E, E,, t ) is given by
Analytic Treatment of Proton Induced Electromagnetic Cascades
365
withgd(s)= an(s)--o~,(s), El(z)is the exponential integral. The expressionfory,(E, E,, t )
is similar, only Vy(s),V$(s)and V$(s)are different from VY,, V?& Vg. Vp, V;, V;, VT, V$
and VP, are given in Appendix A. I n Figs. 7 and 8 F1(E,E,, t ) and P,(E, E,, t ) are shown
as functions of t for different values of E,,/E.
I n Figs. 9 and 10 the ratios Tl(E,
E,, t)/T,(E,E,, t ) and Pl(E, E,, t)/P,(E,E,, t)
are represented as functions o f t for E,/E = lo6.
3.3.2. C o n v e r g e n c e
For which energies are the series
2 (~ Ac o Ts e *
I/n(s, t )
2 (&)”.
and
gn(s,t ) convergent 8
n=O
n=O
W
To answer this question we niust first find the energy region in which
2 (-
Bd/Ecos @ n .
n=o
t ) is convergent.
The integration of (3.8d) yields
St,(s,
(3.12)
(3.13)
,%;,(s,
t) is obviously bounded.
W
We conclude that
2 (-B,/E
-
cos Qn ?&(s, t ) is convergent for I B,/E cos 61 < 1.
n=O
W
Returning t o the convergence of
2 (--Bd/Ecos t9)n - ??,(s, t ) we
recall that f,(s, t ) is
?C=O
given by
x 5in(s,t ‘ ) at‘.
(3.14)
For M r ( s ) and Mg(s) see Appendix A.
1
g,,O(S
+ n) = J d + * - l *g,,o(u)
du.
0
We assume gnno(u)
to be bounded in the region 0 <_ u
Il/,(s, t )I
5
1; then we get
5 2 const (s + n + 1)-1- (s + n)-l- niax (l?io(s, t)l)/An
x ( \ M l ( s + n)I 1 eAl(8+n)t- 11 + 1 M Y ( S+ n)I 1 eA*(s+n)t - 1I}
(3.15)
S. BARMO
366
with A1(s
(3.15)
+ n ) < 0 and I,(s + n ) < 0 for s > 0 and
17w(s,t ) 1 5 2
*
const * (s
?2
2 1 [3] we obtqin instead of
+ n)-l*(s + rL + 1)-l
*
niax (J%,(s, t)I)
x inax (I M ~ ( RI, 1)M<(s)I).
m
This means that
2 (-
B,/E cos O ) , . Yn(s,t ) is convergent for 1 R,/E cos 8 I 2 1.
n=O
m
We can similarly prove that
2 (-Bd/E
cos 0 ) 9 1$,(s,
t ) is convergent in the same
n=O
region.
3.4. Energy Conservation
The average energy associated wieh particles of type i is given by
m
( E i ) = J E i ( E ,t ) dE
-
= iX(1, t )
(3.16)
0
where i ( E ,t ) is the average nuniber of particles of type i and energy (h',E
m
~ - V ( ( St ), =
+ dE),
$ i ( E ,t ) ESd E .
0
To analyse the energy conservation we must account for the K-mesons which are
also produced if a nucleon or a n-ineson interacts with a nucleus.
The energy conservation gives the two following conditions for the functions &( 1)
(i = n,no,K ) and
(i = no,z, K )
sni(l)
(3.17 a)
(3.17b)
Using the conditions (3.17a) and (3.17b) we prove that our solutions (3.6a) and (3.6b)
are consistent with energy conservation. The energy conservation requires (see (3.16))
(&AL 4 + %%f(1, 4 + Y d J ,
t)
+G d I , t)+
%M(L
t ) ) = &.
(3.18)
We still need an expression for the K-mesons. We neglect the energy flowing into the
electromagnetic cascade via kaons. The energy loss by kaon production is then obtained
from
fiM(s,t ) is the solution of (3.4a) and ?cM(s, t ) is given by (3.13), ClIf(s,t ) = ES,
Obviously %Jf(s,t ) and iiM(s,t ) tend to zero for t -+ 00. I n this case we have instead
of (3.18)
hn
t+ m
(&(L
4 + P M U > t ) + r dI
>
t ) ) = EO
using the condition (3.17a) and (3.17h) and the relations [3]
I,(l)= 0, A(1) + B(1) = -I&).
(3.20)
4. Discussion and Conclusions
4.1. noInduced Cascade
The only difference between the solution for a photon induced and a no induced
cascade is the factor 2(s + l)-l. InFig. 1 we see that the solution of a no induced cas-
Fig. 1 The average number of photons r ( E , E,,, t ) with energy greater than E as
function of t for different values of E,/E
The no induced cascade (solid line) in comparison with a photon induced cascade
(dashed curve)
cade is higher than the solution of a photon induced cascade for t < t,,,, . For t > t,,ar
the opposite is true. We assumed that pion and photon had the same energy E,.
4.2. Proton Induced Cascade with no Decay Neglected
We get as result of the scaling law that r ( E , E,, t ) and P ( E , E,, t ) depend only on the
ratio E,/E.
I n Figs. 2 and 3 r ( E , E,, t ) and P(E,E,, t ) are represented as fiinctions of t for different values of E,IE. They are divided into two parts, one part has its origin in the no’s
produced by nuclens and the other one has its origin in the no’s produced by nL’s;they
are called N-part and n-part.
Obviously P(t)and F(t) go to zero for t --f 0. This is consistent with our assumption
that only a proton of energy E, is present a t t = 0.
368
S. Bmnio
As seen from the Figs., the ratio (N-partln-part) is increasing with increasing t.
I n Fig. 2 the ratio takes the value 0.475 for t = 5X0 and 1.15 for t = 25X0. Thus no’s
production by nucleons is more important a t the beginning of the cascade. This is simply
due to the fact that all charged pions are secondaries.
Another effect in the same direction is the somewhat larger attenuation length of
pions (A,/A, = 115/95). These two effects ensure that no’s are produced by nf’s when
t 9 lX,. Therefore the n*’s contribute to the energy of the electromagnetic cascade
also a t these t values. If we compare Pigs. 1 and 2 we see that r(t)
for a proton induced
cascade falls much more slowly than for a photon induced cascade. This is due to the
Fig. 2 The average number of photons T(E,E,, t ) for a proton induced electromagnetic
cascade
T is divided into n-part (dashed curve) and nucleon part (solid curve). The z* decay is
neglected
Fig. 3 The average number of electrons P(E,E,, t ) for a proton induced electromagetic
cascade
P is divided into n-part (dashed curve) and nucleon part (solid curve). The nf decay is
neglected
r 4I
e-+
-
074
I
tfAn1
Fig. 4 The average energy of the electromagnetic cascade calculated with (3.16). The
energy is divided into photon and electron parts
Analytic Treatment of Proton Induced Electromagnetic Cascades
369
Fig. 5 Comparison of T(6,
ED,t ) for two different cascades, a photon induced cascade
with initial energy 0.416, (dashed curve) and a proton induced el~ctromagneticcascade
with initial energy E,
Fig. G Comparison of P ( d , E@,t ) for different cascades, a photon induced cascade with
initial energy O.ilED(dashed curve) and a proton induced electromagnetic cascade with
initial energy Eo
fact that for t
I, no's are still produced which conipensates partly for the loss. I n
Fig. 4 the average energy of the electromagnetic cascade calculated with (3.16) is represented as function of t. A t t = 11.4, (nearly sea level) the energy of the photons is
about 0.23E0 and of the electrons 0.18Eo.
I t seenis to us that these values are small because formula (3.20) says that the
remaining energy is used to produce K-mesons (K-mesons and antiprotons).
This seems 60 be due to the parametrization of the functions yii(u) of ref. [8], which
required a certain extrapolation of the experimental data [ 131.
I n Figs. 5 and 6 a photon induced cascade with initial energy O.41E0 is compared
with the proton induced electromagnetic cascade. It is striking that
1. up to t m 12- 15X, the solutions of a photon induced cascade are higher and that
later the opposite is the case;
2. the rate of decrease of the solutions of a photon induced cascade is greater.
The first remark may be explained by the fact that a t the beginning of a photon
induced cascade the whole energy (0.41E0)is used to produce photons and electrons.
I n the case of a proton induced cascade the energy is supplied slowly and the loss of
electrons and photons is partly compensated (weaker decreasing). The results are obtained by using Table 1.
1.3. The Effects of n k Decay
We consider Tl(E,E,, t) and P,(E, E,,, t) as the first order correction to ro(E,E,, t )
and P,(E, E,, t). I n Figs. 7 and 8,
and P, are shown as functions of t for different values of E,/E. Obviously they are small incomparisonwithT, and Po. The ratiosTl(E, E,, t ) /
To@,E,, t ) and Pl(E,E,, t)/P,(E, E,, t ) are represented in Figs. 9 and 10 as function
r,
24
.Inn. Physili. 7. Folge, Bd. 33
S. BARMO
370
of t for Bo/E= lo5. €‘,/Po is smaller thanTl/To and both are increasing with t . They are
increasing with t because the nf’ S contribiiting to no’s production will always have
sinaller energy with increasing t .
PJP,is smaller because photons are produced directly by n*’s and they are therefore more sensitive to their energy. TJT,, and PJPo are of the order of niagnitude of
I 0-3.
Table 1
an
an
an
an
.200
.300
,400
.500
.(i00
.700
.8U0
.900
1.000
.074
.lo1
.125
.145
.I62
.177
,302
.251
.216
.186
.I62
.142
.12,6
.111
.099
.089
.081
,073
-.355 -1.488
-.225
-1.140
-.124
--A83
-.046
-..(i92
.016
--.548
.OG5
-.439
.lo5 -.354
-.288
.137
.163 -.23(i
.184
-.195
.202
- A62
.217
-.13G
- .OG1
.229
-.116
-.053
,240 -.I297
- .047
.249 -.Of33
-.042
.256
- .O71
-.037
.263 --.OGI
- .033
.m - .052
-.030
.274
- .046
-.027
.278
--.I340
- .D24
281
.035
-.022
.285
--.030
-.020
.288
-.O27
-.OM
.a0
- ,024
- .O17
292
- .021
-.oi5
.mi -..oig
- ,014
.296
- .017
- .013
. a s - .015
- .t)l2
.299 -.013
.l!ll
.202
.213
.22d
.231
.238
.245
.25%
.258
,263
.268
.272
.277
.281
281
.288
.291
.294
297
.299
1.100
1.200
1.30O
1.400
1.500
1.G00
1.700
1.m
1.900
2.000
2.100
2.200
2.300
2.400
2.500
2.600
2.700
2.800 .308
w o n .304
3.000 .307
.Of7
.0Gl
.05G
.05l
.047
.044
.041
.038
.035
.033
.031
.n29
.027
.O~G
.024
.023
.0$2
-
I,
an
$
an
-.543
-.423
-.335
--.270
-.220
-.182
-.152
-.128
-.lo9
-.093
--.Of30
--.070
-
Ynn
-
Ynn
-,,
-,
Snn
Sd
Ynn
4.065 1.107 2.122 -4.702 12.843 -.429
9.388 -.326
2.971 .883 1.709 -3.601
2.200
.712 1.392 -2.791
6.951 -.249
5.211 -.190
1.649 .579 1.144 -2.187
1.251 .476 .949 -1.732
3.963 -.14G
3.031 -.I12
.959 .394 .794 -1.386
2.349 -.086
.744 .329 .670 -1.118
1.838 -.066
.569 -.910
.582 .277
1.451 -.05(J
.459 .235 .486 -.746
1.156 -.038
.366 .200 .418 - - . M i
.294 .172
,362 - .513
.928 -.029
.750 -.We2
.237
.148 .315 -.430
.193 .128 .27G - .362
.611 -.016
.501 -.012
.158 A12 .a42 -.306
.413 -.009
.131 .098 .214 --.261
.342 -.007
.lo8 .086 .190 -.223
. ~ S F -.oo5
. I G ~ -.w
.ago .076
239 -.oo&
.076 .om
.m -AX
.20% -.003
.OM .060 .13G - .144
.054 .Oh4
.122
-.125
.171 --.003
.046 .048 .111 - .I10
.l45 - .O03
.039 .044 .loo - . o ~ A
.IN -.003
.034 .040 .091 -.085
.I06 -.003
--.on .ON - . O O ~
.029 . 0 3 ~ .om
.079 - ,004
.025 .033 .076 - .066
.om -..on5
.022 .030 .070
-..059
.059 - .OOCi
.OH
.027
.065 - .053
. o z - .oo7
. o x .0%5 .OGO --.047
.014 .023 .055 - .042
.045 - .007
The numerical values for a,(s), a n ( s ) ,@C(s), a’,(s), a;E(s),a’&), GnTt(s),
ij,,(s), Snn(s),
and gd(s) are given.
We calculated them using the following functions [8]and numerical values
Sk&),i;l,(s)
4
2
g1171( u ) = - - - u ,
3
3
g,,(u)
=
7.2(e-’, f Cgu),y,,(u)
1
- yTtZ(u), gnno(u) = Y g n n ( u ) , d,,
2
’2
=
95 g/cm2, A,
=
24e-i2d,
= 115 g/cm*,
X,
=
qnna(u)=
36.4 g/cm?.
For A,(s) and &(a) we used the Tables of ROW [3]. The fact that A. LILASDused another
scaling law is taken into account. The saddle point method is used to evaluate the
integral expressions giving the solutions.
371
Analytic Treatment of Proton Induced Elect,romagnetic Cascades
Fig. 7
The first correction term Tl(E,
E,, t ) as function of t for different valucs of Eo/E
Fig. 8 The first correction term P,(E, E,, t) as function o f t for different values of E,/E
l
70
20 tfXo]
I5
25)
Fig. 9
Fig. 9 The ratio r l ( E ,E,, t)/T,(E,E,, t ) as
function o f t for E,/E = lo5
+
70
.
-
.
A
I
75
20
"XJ
25*
Fig. 10
Fig. 10 The ratio P l ( E ,E,, t)/P,(E,E,, t ) as fonction of t for Eo/E = lo5
I wish to thank Prof. H. PILKUIIN
for suggesting and encouraging this work and for
many helpful discussions.
Appendix B
Using simple considerations we shall estimate the energy lost by a nucleon t,o produce no's.
Let us assume that the nucleon had an energy Eo ( 9 Bd).After one interaction the
nucleon will have an energy EN = aEo with a < 1. After two interactions its energy will
be azEoand so on.
A t every interaction of the nucleon the energy
and the energy Po E f l is used to produce no's.
pi
*
EN is used to produce z f ' s
7
After n interactions the energy used to produce no's is given by
+
+
+ +
E$ = PoE0 PoxE0 /loc~~Eo
Booin-'E0 = BoEo(1 - an)* ( 1 - a)-'.
Let n, be the multiplicity of n*'s produced in the first interaction of the nucleon (n*'s
first generation). The average energy of each pion will be Ei = P+Eo/nl.
If Ei > Bd,the n*'s of 1. generation will produce z*'s and no's.
6,Ei will be used to produce n*'s.
aOE: will be used to produce no's.
a * *
1 is greater
The new n * ' s will produce other n*'s if their energy E i = 6, '+E~
n1
122
than B, and this process will continue until the energy of each pion is comparable with
Bd
*
Analytic Treatment of Proton Induced Electromagnetic Cascades
373
Froin the first pion generation El,,o is used to produce no’s
=
El,,,
f 60 d+p+EO
= SOf9+EO(l-
+
80
d$@+EO
ST) (1 - S+)-1.
+ +
“’
60
ST-lb+EO,
is the number of interactions, after which the energy of n*’s is comparable with Bd.
Similarly we can calculate the energy of the second n* generation (they are produced
through the second interaction of the nucleon) used to produce no’s.
It is given by
m,
E;,o
= 01
&f9+EO(l - a?) (1 - a+)-’.
We can continue this process and all n*’s contributions can be calculated.
For E, = 106 GeV, S+ = 0.6, @+ = 0.26, oi = 0.5, So = S + / 2 and Po = @+/2 we calculate the average energy used to produce no’s.
For the niultiplicity we use El21
-
(z,) = 2 log El
+ 1.6,
(n8>is the multiplicity of produced charged particles.
At t M 6An, about .46E, is used to produce no’s.
For Bd--f 0 this part becomes 2Eo/3.
We conclude that a relatively large part of the energy is used to produce no’s.
References
[l] S. HAYAKAWA,
Cosmic Rays Physics, Wiley-Interscience (1969).
and W. HEITLER,Proc. Roy SOC.Lond. A 159, 432 (1937).
[2] H. J. BHABRA
[3] B. ROW, High Energy Particles, Prentice Hall Inc. 1952.
[4] G. MOLIERE and P. BunINr in W. HEISENBERG
,,Kosmische Strahlung“. p. 365. Springer-Verlag,
Berlin 1953.
Phys. Rev. Lett. 23, 1415 (1969).
[5] R. P. FEYNMAN,
[C,] J. BENECKE,
T. T. CHOW,
C. N. YANGand E. YEN. Phys. Rev. 188, 2159 (1969).
[7] A. LILAND“Discussion of High Energy Cosmic Ray Muon Spectra in Terms of Primary Spectra
and Muon Production Spectra”. Diplomarbeit, Universitat Karlsruhe 1971.
[8] A. LILAND“Analytic Evaluation of Pion, Kaon and Muon Distribution in the Atmosphere”,
Doktorarbeit, Universitat Karlsruhe 1973.
[9] H. BETHEand W. HEITLER,Proc. Roy SOC.Lond. A 46 (1934).
Cosmic Gamma Rays, Mono Book Corp., Baltimore 1971.
[lo] F. W. STECKER,
[ l l ] H. PILKUHN
in “Lectures on Space Physics I”, p. 1-11Bertelsmann Universit&tsverlag,Giitersloh
1973.
[12] E. L. FEINBERG, Phys. Rep. 5 c, 240 (1972).
[13] D. HAIDT,Particles Production From Nuclei a t 24 GeV/c, CERK, TCL Int. 71-11 (1971).
Bei der Redaktion eingegangen am 16. Oktober 1975.
Inst. f. Theoret. Kernphysik d. Univ.
Ansrhr. d. Verf. : Dr. S. BARMO,
D-7500 Karlsruhe 1, Kaiserstr. 12
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