104
Supplement of the Progress of Theoretical Physics, No. 32, 1964
High Energy Gamma-Rays in the Atmosphere
and Muons Underground and Underwater
Satio HAYAKAWA, Jun NISHIMURA* and Yoshiaki YAMAMOTO
Department of Physics, Nagoya University, Nagoya
*Institute for Nuclear .Study, University of Tokyo, Tokyo
The behaviour of high energy nuclear active particles, ')"-rays, muons and neutrinos.
in the atmosphere aru:l underground is analyzed under the light of recent experimental
results obtained by several groups in Japan. Through our analysis the following
conclusions are obtained. (1) The attenuation length of theN-component of energies
above a few TeV is about 110 g cm- 2, as derived from the altitude variations of the
')"-ray flux and the nuclear interaction frequency. (2) The break in the 7-ray spectrum
can be attributed to that in the primary spectrum, implying that the features of nuclear
interactions do not appreciably change as energy increases as high as 100 TeV.
(3) The intensity-depth relation of the muon flux is found to be consistent with the
')"-ray spectrum by taking proper account of the range fluctuations of muons. ( 4) The
ratio of kaons to pions at high energies is consistent with that observed in jet showers.
The present paper is such a review article that contains a summary of experimental
data, the method of dealing with cascade showers, the mathematical treatment of the
range fluctuations and the neutrino fluxes expected.
§1. Introduction and historical survey
The present article is a modern version of an earlier review on high
energy cosmic ray phenomena written in 1950 by one of the authors
(S. H.) ,1) which is referred to as I. Comparing I with the ,present one,
one would find rather minor differences between these two, in spite of
the fact that great effort for experimental work has been made in the
last fifteen years.
The previous article, I, was based on a series of theoretical works
developed during 1947-1950 by a Japanese group, in which two of the
present authors (S. H. and J. N.) were involved. It was intended to draw
a consistent picture of high energy cosmic ray phenomena on the basis of
the two-meson theory and the theory of electromagnetic interactions, and
to clarify the properties of nuclear interactions. It was the time when
artificially produced mesons began to be available and the type of the rrmeson was one of the central problems to be investigated. One was also
wandering, if the interaction of the rr-meson was as strong as the strong
interaction and if the interaction of the ,u-meson was as weak as the weak
interaction.
In comparison with these questioned problems, the two points which
provided the basis of our analysis in I were considered as established.
High Energy Gamma-Rays in the Atmosphere
105
The two-meson theory proposed by Sakata, Tanikawa and Inoue in 1943
was confirmed by the direct observation of the TC- and tt-mesons in photographic emulsion in 1947. The lifetime of the TC-meson decaying into the
tt-meson was estimated by Hayakawa2> through the analysis of the intensitydepth relation of cosmic rays underground. The intensity underground had
been known to decrease in proportion to a negative power of depth, the
absolute value of the power index increasing gradually beyond a depth of
about 500 m water equivalent ( w. e.). The evidence for the bending of
the intensity-depth curve was beyond doubt due to the cosmic ray observation by Miyazaki 3> during the War in Shimizu tunnel, the deepest point
until 1960. This fact had been interpreted as due to the increasing energy
loss rate, even when the two-meson theory was taken for granted. 4) Much
earlier in 1941, however, Tomonaga showed in his unpublished work (the
result being published in a revised form in reference 5) that the proper
application of the theory of electromagnetic interactions could not predict
the bending as observed. In fact, the bending was accounted for in terms
of the TC-tt decay, as was pointed out independently also by Greisen. 6>
In the TC- p. decay neutrinos are also produc~d. Estimating the neutrino
flux underground, Hayakawa, Machida and Umezawa 7> discussed the observability of neutrino interactions.
Once the properties of underground cosmic rays were understood, the
energy spectrum of p.-mesons and consequently the production spectrum of
charged TC-mesons were derived. The ·latter is directly connected with the
production spectrum of neutral TC-mesons and consequently with that of
their decay r-rays. Through this genetic relation, Fujimoto, Hayakawa
and Nishimura8> discussed the behaviour of the electronic component
arising from these r-rays in the atmosphere. They demonstrated that the
major part of the electronic component at high altitudes was accounted
for in this term and its intensity was essentially parallel to that of the
nucleon component, which was also shown by Rossi. 9 > The nucleon component consists of nucleons and charged TC-mesons, which are subject to
the nuclear cascade process. This was discussed in connection with the
tt-mesons and the electronic component by Hayakawa and Nishimura 10> and
also independently by Heider and Janossy.ll)
The present article covers the problems described above, but does not
contain several subjects discussed in I, such as the interactions of tt-mesons
and the structure of extensive air showers. On the other hand, there are
a number of new points, in addition to that discussions in the present
article are more quantitative than before. Firstly, the gradual bending of
the energy spectrum of the electronic component is taken into account.
Secondly, the range fluctuations of tt-mesons underground are discussed.
Thirdly, the contribution of K-mesons is considered These new points
106
S. Hayakawa, J. Nishimura and Y. Yamamoto
come from the development of cosmic ray research in recent years; the
last point is obviously new, because the name of the K-meson did not
present in 1950.
The first point comes from a series of works by Japanese emulsion
chamber group, as described in an accompanying paper,t2> referred to as
II. The bending of the r-ray spectrum at about 1 Te V was first discovered
when they exposed a large emulsion chamber at Mt. Norikura in 1958.13>
The change of the power index was found to be so large as unity that
this was suspected to be due to the change in the characteristics of nuclear
interactions. Further investigations have shown, as described in II, that
families of r-rays are occasionally observed and, if they are properly taken
into account for the correction of the r-ray spectrum, the change of the
power index is not so large as was thought before.l4) Alternatively this
has been interpreted as due probably to that of the primary spectrum/ 5>
as noticed from air shower evidences.
Thus the emulsion chamber
experiment is found to be not too directly responsible for elementary
interactions, but to be as indirectly as air shower experiments; the family
of r-rays is nothing but the very initial stage of an air shower.
On the other hand, the r-ray spectrum is very important for the
morphological study of cosmic rays. Its relation to the primary spectrum
is a typical example and is discussed in §3 and §4. The relation to the
muon intensity is another example. When the r-ray spectrum above 1Te V
was observed as very steep, one suspected that the muon intensity observed
by Miyazaki 3> could not be consistent with the r-ray intensity, if both arise
mainly from pion decays. However, the muon intensity observed by
Barton16) at about the same depth as Miyazaki's gave a much lower value
which would have been consistent with the steep r-ray spectrum. On the
contrary, the burst size spectrum observed by Osaka City University and
Institute for Nuclear Study groups 17 > gave a high muon intensity in the
same energy range. More recently the Japan-Indian collaboration work 18>
at Kolar Gold Field in India has obtained the muon intensity down to
8400 m w.e., and the intensity-depth curve obtained by them has been
found to lie between Miyazaki's and Barton's intensities.
In order to compare the intensities of underground muons and atmospheric r-rays, one has to know the energy loss rate of muons. Below 0.5
Te V the ionization energy loss is a dominant process, and the loss rates
by bremsstrahlung and direct pair creation are of minor importance. In
a relatively low energy region, an experimental cheque of the energy loss
rate is possible by taking account of the momentum spectrum of muons
measured with a magnet spectrograph. 19> An accurate measurement of
muons underwater has been carried out by Osaka City University group, 20>
thus verifying the theoretical energy loss rate, in particular the energy
High Energy Gamma-Rays in the Atmosphere
107
loss rate due to the direct pair creation calculated by Murota, Ueda and
Tanaka. 21 >
Above 1 Te V bremsstrahlung and direct pair creation are dominant
processes, and the energy loss rate thereby is essentially proportional to
energy. The latter results in that the fluctuations in energy loss are of
primary importance in the intensity-depth relation of muons. This problem
has been treated numerically by Miyake 22 > and analytically by one of the
authors (J. N.) .23) Mathematical details of the latter are given in Appendix
D.
Through such analyses the comparison between the r-ray spectrum and
the muon intensity becomes possible, and they are shown to be consistent
with each other, provided that the contribution of kaons is not greater
than that of pions. This K/ rc ratio is consistent with the one found for
jets observed in emulsion24 ) as well as with the positive-negative ratio of
muons at sea level,25> although the energy region in the latter is much
smaller than that of our interest.
Another important consequence of the muon observation at Kolar
Gold Field is the demonstration that the neutrino experiment is feasible
at such a great depth, on account of that no particle was detected during
two months at 8400 m w.e. 26) This is good enough to put the lower limit
of the weak intermediate boson mass as large as that obtained with accelerators. Being stimulated by this experiment, a number of attempts of
observing neutrino interactions are under way. For the purpose of reference
we give the neutrino flux expected in §6.
The present review is based on a series of systematic works developed
in Japan, as described above. However, this is not entirely a review of
published works, but contains some new points in the following three
respects. Firstly, the bending of the r-ray spectrum is accounted for as
due to that of the primary spectrum, secondly, the muon spectrum is
derived from the intensity-depth relation of muons on account of the range
fluctuations of muons, and thirdly, it is compared with the r-ray spectrum
with particular attention to the K/rc ratio. In addition, discussions of
various problems concerned are made in rather detail for the sake of
completeness and of convenience of the readers who are not always familiar
to cosmic ray physics.
§2. Summary of experimental data
Although we are mainly interested in the energy region higher than
1 Te V, we add some experimental data concerning those particles which
have energies below 1 Te V but higher than 100 GeV, because discussions
will be made on the energy dependence of the spectral index which is
S. Hayakawa,
108
J.
Nishimura and Y. Yamamoto
rather pronounced around 1 Te V. The experimental results on r-rays
(electrons inclusive) are discussed in detail in II and are, therefore,
mentioned only briefly for the sake of completeness.
2 · 1.
Electronic component*)
In most of experiments both r-rays and electrons are observed without
discrimination. It is therefore convenient to put them together as the
electronic (E) component. Observations made by several groups and
their experimental conditions are listed in Table I.
The integral energy spectra of the E-component observed thereby are
shown in Fig. 1. They are expressed approximately by power shapes;
the power indices obtained are given in Table I. It is seen that each
spectrum. cannot exactly be represented by a single power law but shows
gradual steepening as energy increases. It may, therefore, be appropriate
to express the observed spectrum as consisting of two parts with different
spectral indices, one below 1 Te V and the other above 1 Te V, as long as
the energy range observed is wide enough. These spectra are not always
consistent with each other; possible reasons for the discrepancy are discussed in II. According to II, the following values of the spectral indices
are considered as most reliable:
{1=1.8---2.0 below 1 TeV,
{1=2.2----2.3 above 1 TeV.
(2·1)
The absolute flux of the E-component and its altitude variation are
shown in Fig. 2. For the experiments at very high altitudes, the depths
are corrected by taking account of the zenith angle distributions. The
intensity-depth curve has an exponential form at great depths and a peak
characteristic to the cascade shower at about 100 g cm- 2 • The attenuation
length in the exponential part is estimated as
L= 110± 10 g cm- 2 •
(2·2)
This is consistent with that derived from the zenith angle dependence as
well as the production rate of the E-component.
2·2.
Muons
The energy spectrum of muons below 100 GeV has been measured
directly by means of magnet spectrographs. 26 ) Above 100 Ge V the association of electron showers with a muon is such a disturbing source of
experimental bias, that other methods are required to obtain a reliable
spectrum, although this method can be extended to 1 Ge V with suitable
correction. From 100 Ge V to several Te V the burst size spectrum is
*)
For details, see an accompanied paper, referred to as II, by Japanese emulsion group.
High Energy Gamma-Rays in the Atmosphere
Table I.
Atmospheric
depth
(g cm- 2 )
Apparatus
9 (26)*
balloon
em. stack
balloon
em. chamber
balloon
em. chamber
airplane
ion chamber
airplane
em. stack
airplane
ion chamber
Chacaltaya
em. chamber
Norikura
em. chamber
22 (37)
30 (57)
197
220
310
550
730
730
1030
I
I
"'T
cl\lr
ion
sea
10M-
1
:1
109
Observations on 'Y-ray energy spectrum.
Energy
range
(TeV)
Exponent
of
the power
Energy
range of
.EEr (TeV)
Exponent
of .EEr
Reference
0.1-2
1 9+0.3
. -0.2
0.3-5
1. 75±0. 20
0.1-2
2.0±0.2
0.03-2
1. 76±0.11
0.1-5
1. 92±0.14
Moscowd)
0.3-2
2-10
0.03-2
2. 3±0. 2
2.8±0.3
1. 83±0.13
0.3-2
2-10
0.1-5
2.5±0.2
3.5±0.3
2.13±0.17
Bristole)
0.5-10
2.2±0.15
1.2-8
2.0±0.5
0. 35-1
1-10
lo 002-0.02
2.0±0.2
2.3±0.2
1.82
IO 002-0.02
1. 86
Chicago~'~>
0.3-10
2.04±0.22
Bristol'Bombayb)
Japan°>
2-23
2.1±0. 3
Moscow f)
JapanBrazilg)
Japan h)
Kobel)
'.1:
I
Kobel)
'1:
lOll
*The values in parentheses represent the effective depths.
]. M. Kidd, Nuovo Cim. 27 (1962), 57.
F. Abraham, ]. Kidd, M. Koshiba, R. Levi Setti, C. H. Tsao and W. Wolter, Nuovo Cim.
28 (1963), 221.
b) P. H. Fowler, Proceedings of the International Conference on Cosmic Rays, at Jaipur
5 (1963), 182.
c) 0. Minakawa, Y. Nishimura, M. Tsuzuki, H. Yamanouchi, H. Aizu, H. Hasegawa, Y. Ishii,
S. Tokunaga, Y. Fujimoto, S. Hasegawa, J. Nishimura, K. Niu, K. Nishikawa, K. Imaeda
and M. Kazuno, Nuovo Cim. Suppl. 8 (1958), 761.
d) L. T. Baradzei, V. I. Rubstov, Y. A. Smorodin, M. V. Solovyev and B. V. Tolkashev,
]. Phys. Soc. Japan 17 (1961) Suppl. A-III, 433. Proceedings of the International Conference on Cosmic Rays, at Jaipur 5 (1963), 283.
e) ]. Duthie, P. H. Fowler, A. Kaddoura, D. H. Perkins and K. Pinkau, Nuovo Cim. 24
( 1962), 122.
f) Same as d).
g) M. Akashi, K. Shimizu, Z. Watanabe, ]. Nishimura, K. Niu, N. Ogita, Y. Tsuneoka, T.
Taira, T. Ogata, A. Misaki, I. Mito, Y. Oyama, S. Tokunaga, A. Nishio, S. Dake, K.
Yokoi, Y. Fujimoto, T. Suzuki, C. M. G. Lattes, G. Q. Orsini, I. G. Pacca, M. T. Cruz,
E. Kuno and S. Hasegawa, Proceedings of the International Conference on Cosmic Rays,
at Jaipur 5 (1963), 326.
h) reference 12).
i) T. Kameda, J. Phys, Soc. Japan 15 (1960), 1175.
T. Kameda and T. Maeda, J. Phys. Soc. Japan 15 (1960), 1367.
j) C. A. Randal and W. E. Hazen, Phys. Rev. 81 (1951), 144.
k) B. V. Sreekantan and S. Naranan, Proc. Ind. Acad. Sci. 3.6 (1952), 97.
l) L. M. Bollinger, Ph. D. Thesis, Cornell University (1951).
m) S. Ya. Babetsky, A. Z. Buya, N. L. Grigorov, E. S. Lotskevich, E. I. Massalsky, A. A.
Oles and V. I. Shestoperov, J. Phys. of Acac. Sci. USSR 40 (1961), 1551.
n) H. P. Babajan, S. Ya. Babetsky, I. I. Boyagan, Z. A. Buya, N. L. Grigorov, E. S. Lotskevich, A. A. Mamigan, E. I. Massalsky, A. A. Oles, V. I. Shestoperov and Ch. A.
Tretyakova, Proc. of Acad. Sci. USSR 26 (1962), 1559.
a)
110
J. Nishimura and Y. Yamamoto
S. Hayakawa,
/cm 2 sec sr
-4
J.
10
"'
"'
-5
10
•
••
6
16
*c
.i.
D~
"•
·~·•
'~I
7
10
r~J.
~L
-8
10
~f.
t \ ++.
t
~
•
0
'
•
+
9
90 +
-I 0
10
?
T
t+ ~
tl+
II
?
0
t
10
11
Fig. 1. '}'-ray spectra observed at the various altitudes.
I
e;
I
I
.A;
I
Chicago,a>
Moscow/)
I
.;
Japanese balloon,c)
)!<; Bristole)
Mt. Chacaltaya,g)
0; Mt. Norikurah)
I
I
l
References are the same as in Table I.
available for obtaining the muon spectrum. 17)' 27) However, the size of the
burst produced by a muon of given energy fluctuates so much that the
muon spectrum can not be obtained from the burst size spectrum in a
straightforward way. We have, therefore, only to mention here that the
burst size spectrum gives a muon spectrum not inconsistent with that
111
High Energy Gamma-Rays tn the Atmosphere
200
400
600
800
1000
-2
gem
Fig. 2. Altitude variations of the -r-ray fluxes at energies 0.3, 1.0, 3.0 and 10 TeV.
The full curves represent the cascade curves for L=lOO g cm- 2 and dashed curves
for L = 110 g cm- 2 • Signs are the same as in Fig. 1.
obtained with other methods.
Covering the whole energy range, in part of which the above two
methods are applicable, the intensity-depth relation gives the muon spectrum. The intensity of muons against the depth, which is customarily
measured in units of meter water equivalent (m.w.e.), has been measured
by many authors. Down to 1800 m.w.e. many experimental points are
available, 29),so) and a smooth curve can be drawn rather easily through
S. Hayakawa,
112
J. Nishimura and Y. Yamamoto
them. At about 3000 m.w.e. observations in a tunnels> and under water16>
gave considerably different intensities. The intensities derived from the
measurement of inclined muons are closer to the former. 3n A recent
measurement of the vertical intensities down to 8400 m.w.e. has given
values lying in between. 18> It is worth while to note that no count was
obtained with detector of an area of 1.6 m 2 in a period of 2800 hrs at the
deepest point. These data on underground muons are reproduced in Fig. 3.
In deriving the depth in m.w.e., most of underground experiments
are suffered from the roughness of the ground surface and the density of
rock above the observing depth, which are not always known with sufficient accuracy. A recent underwater experimene2> shown in Fig. 4 has
shown that in most of underrock experiments the rock density seems to
have been slightly overestimated, so that the energy scale may have to be
reduced by about 10%.
In order to convert the intensity-depth relation to the energy spectrum,
we have to know the energy loss rate. This is conventionally expressed
as
e;
/cm 2 sec sr
I(
.
Jf
10
3
\7;
....
•
4
10
10
COR;
-...•.
.;
.A;
Miyake et al./ 8 ) (!)R; Randal et aU>
~; Sreekantan et al.k>
Barett et al., 31 )
OM; Miyazaki 3 )
Bollinger,l>
Clay et al., 30 >
X; Wilson 29 >
Barton, 16 )
.•
...
"'\4.
5
/cm2 sec sr
..•
I
I•
J~\
;6
\tOR
16
-7
10
10
0
\M
~e
10
.r=4
r=~
;.Kl
10
Fig. 3.
10\Z
0
.
~.
-5
~
\)
10
-
D
'~
:!3
10
• Normalization point
-3
)3
IOmwe
Vertical intensity of muons versus
depth in Rock.
10
~
~.
-6
10
<
~
-7
10
-8
10
10
Fig. 4.
100
1000
mwe
Vertical intensity of muons versus
depth in sea water obtained by
Higashi et al.2°>
High Energy Gamma·Rays in the Atmosphere
113
dE
E~ 2 ) ,
- =a+bE+c In ( -~dx
(2·3)
mf.J.c
where mf.J. is the muon mass and E~ is defined by (D · 2). The values of
these parameters, a, b and c, are evaluated for rock and for sea water, as
given in Appendix D.
The values of a, band c depend on the medium as shown in Appendix
D, through which muons penetrate, because the energy loss rate depends
weakly on the average atomic
number Z, the average mass
number A and the density p.
Hence the energy-range relations in rock and sea water are
-5
different, and they have to be
10
used for transforming the intensity-depth relation to the energy
spectrum.
In Fig. 5 the energy spectrum of muons is shown by
7
referring to underrock 18> and
10
20
underwater > experiments. In
\
deriving the spectrum above 1
Te V, the fluctuations in the
energy loss are of primary im ·
portance, because bremsstrahlung and direct pair creation
9
10
are two main loss processes and
the fraction of energy lost thereby is large. The range fluctu-I c
ations of muons underground
10
have been treated numerically22>'32> as well as analytically. 23)
-I I
Referring to the latter, the
10
intensity-depth curves of muons
expected for several power
energy spectra of muons are
drawn in Fig. 3. It can be
seen that the best fit with the
Fig. 5. Integral energy spectrum of muons.
observed one by Miyake et aP8>
(!); Miyake et al./ 8) 0; Higashi et al. 20 )
is obtained for a power index
X ; W olfendale et a}_19)
of the integral energy spectrum of
\
\
\
~
\
\\
\
\
r
3.3,
(2·4)
114
S. Hayakawa, J. Nishimura and Y. Yamamoto
provided that we adopt the value of a, b and c and take the energy
dependence of b into account, as discussed in Appendix D.
In this way one can deduce the energy spectrum of muons up to 10
Te V from the intensity-depth relation. However, the following remark
may be necessary. The theoretical values of a, b and c, in particular the
energy loss rate by direct pair creation, are not free from some ambiguity
which is inevitably associated with the accuracy of calculations. Moreover,
the properties of rock all the way along the paths of muons and consequently the depth scale may also be subject to some uncertainty. In order
to avoid these possible errors, the underrock intensities are compared with·
the underwater intensities20) and the momentum spectrum measured with
a magnet spectrograph. 19) Since the underwater observation is, in principle,
regarded as free from these errors and it has given a momentum spectrum
consistent with the magnet spectrograph data, we normalize the momentum
spectrum obtained from the underrock data to the underwater spectrum.
This results in the reduction by 10% of the energy scale applied for the
transformation of the underrock data to the momentum spectrum by using
the values of a, b and c given in Appendix D. The momentum spectrum
drawn by a full line in Fig. 5 is obtained in this way and is regarded as
the most likely momentum spectrum of muons at sea level.
The procedure above implies that the energy loss due to the nuclear
interactions through the electromagnetic field of the muon is not so large
as to affect the conclusion. This gives the upper limit of the contribution
of photonuclear interactions to b and consequently that of the photonuclear
cross section as
(2·5)
2 · 3.
N-component
Nuclear active particles are detected through their production of
secondary particles, particularly through large cascade showers arising from
secondary neutral pions. The total energy converted to r-rays, .SEr, is
estimated by means of the emulsion chamber technique described in II as
well as of the ionization pulses occuring in ionization chambers. The
analysis of jets observed with large emulsion stacks24) tells us that .SEr is
proportional to the total energy given to all secondary charged particles,
Ech, as
.SEr/Ech = 1/2,
(2·6)
and that the fraction of energy given to all charged particles by a primary
nuclear active particle of energy Eo is
(2·7)
High Energy Gamma-Rays tn the Atmosphere
115
for E 0 =1,..._,10 TeV.
The inelasticity coefficient, Kch, given in (2 · 7) is the average value with
respect to the distribution f(Kch). The same is the case for Ky, the fraction
of energy given to all r-rays. Its average value is
(2·8)
In the comparison between the energy spectra of the N-component and
.2Ey, the energy scale of the former is obtained from that of the latter by
multiplying the latter by a factor
(2· 9)
<I/ Ky)-- 1 is definitely different from
obtained, if the inelasticity distribution
The distribution has been measured
of energies between 20 and 100 Ge V
cop per as83)
f(Ky)ooexp[ -A/(1
Ky) 2] ,
2
/m yr
(2 ·10)
where A is constant. This
is consistent with f(Kch)
measured by means of a
calorimeter for higher energy
nuclear active particles. 84) If
(2 ·10) is assumed to hold at
still higher energies, we may
use the energy scaling factor
<I/Ky)~9.5
for
<Ky), and the difference can be
j(Ky) is known.
for primary nuclear active particles
in the collisions with carbon and
\
6=2.1 ±0.3
I
10
\
a~2.
\
(2 ·11)
The spectrum of .2Ey
given in II enables us to
deduce that of nuclear active
particles by using the energy
scaling factor (2 ·11). The
latter thus obtained is shown
in Fig. 6. This is represented
by a power shape with the
spectral index
\
1d 4
EN=9.5
Fig. 6.
a
2.1
0.3.
(2 ·12)
I\
1d5
~
eV
rEr
Integral energy spectrum of nuclear
active particles at Mt. Norikura.
S. Hayakawa, J Nishimura and Y. Yamamoto
116
By reference to other experimental data, the altitude variation of the
N-component intensity can be obtained from the frequency of r-ray families
with 2E'Y greater than 2 TeV.
The altitude dependence is re/crn sec sr
10
presented by an exponential form
I
a
I
I
with the attenuation length equal
I
to that given in (2 · 2), as shown
I
Jr'I
in Fig. 7.
~;b
I
2
!
I
~
2 · 4.
I
~L"II0gcm'
Primary energy spectrum,
IU
Extrapolating the N-component spectrum to the top of
the atmosphere, we would be
td
able to obtain the primary spectrum. In view of the inaccurate
knowledge about the former,
JcJC
however, the spectra of r-rays
and muons are also employed
for this purpose. Thus the deri6()0
400
eoo
1000
200
vation of the primary spectrum
would consist in an important
Fig. 7. Altitude variation of N-component flux
at 2 TeV.
part of our studies. Several at(a)
Air top, 36 )
(b) Moscow/ in Table I)
tempts have been made for
(c) Bristol, e ln 1'able I)
deriving the primary energy
(d) Mt. Chacaltaya,g In Table I)
spectrum from the energy spectra
(e) Soviet,rn>
(f) Mt. Norikura, 12 )
of r-rays, muons and theN-com(g) Soviet,n)
ponent.35-sr> Although they are
different in detail, the methods of derivation adopted thereby are essentially
the same. Thus they agree in the point that the primary spectrum can
be represented by a single power law between 10 GeV and 100 Te V; it
may be expressed simply as
~~~
'\0
""'
1
Fp(E)
1 X E-1.6 cm- 2 sec- 1 sr- 1
(2 ·13)
for 10 GeV<E<3 x 104 GeV,
where E is in GeV. The spectral index and the coefficient may have
uncertainties of
0.05 and of 50%, respectively.
At high energies the spectrum is estimated from extensive air showers
Fp(E) = (3.2 + 0.5) X 10-10 (E/10 6)
0 1
2
- .1± '
cm- 2 sec- 1 sC1
(2·14)
for 10 GeV <E<4 X 10 GeV.
6
8
Connecting (2 ·13) and (2 ·14) smoothly, we gtve the energy spectrum of
High Energy Gamma-Rays zn the Atmosphere
-2
10
117
'~
..,.~
%~
?o
'0~
I
~
I
I
•
~ i\.
~
:
I
I
\
'\.
~~
~
·10
10
\
0.·
1\~
\0~
I
-12
10
\ \.00:0
\
-14
10
\
1\
\
I
-IS
10
\
I
!
I
Fig. 8.
Primary nucleon spectrum.
primary nucleons in Fig. 8.
Although the spectrum in Fig. 8 may be subject to future revisiOn, it
is important to notice the fact that the primary spectrum bends between
1014 eV and 1015 e V. This should result in the bending of the N-component
spectrum in the atmosphere; the break energy is supposed to decrease
with increasing depth. The break should be further transferred to the
spectra of r-rays and muons.
§3. Energy spectrum of nuclear active particles
and its altitude dependence
As is described in §2 · 4, the energy spectrum of primary cosmic rays
may change its slope with increasing energy. The energy at which the
S. Hayakawa,
118
J. Nishimura and Y. Yamamoto
slope changes should decrease as the altitude decreases. In order to illustrate
how this happens, we start with a simplified primary spectrum with a
break at Ec and the spectral indices below and above Ec being a 1 and a 2
respectively. The values of these parameters are chosen by referring to
the primary spectrum adopted in §2 · 4 as
(3·1)
Further simplification is made by assuming a constant elasticity of the
nuclear collision and a constant interaction mean free path. Their values
are chosen respectively as
(=1/2,
l=80gcm- 2 •
(3·2)
Under these simplifying assumptions, the altitude vanatwn of the energy
spectrum of survival nucleons is treated in Appendix C.
The differential energy spectrum thus adopted in (C ·1) has a break
at Ec, as shown in Fig. 9. The break looks rather smooth in the integral
energy spectrum given in (C · 9) and shown in Fig. 10. The break is
/cm 2 sec sr
/cm2 sec sr eV
-23
10
-II
10
Fig. 9. Differential spectra of nucleons at various
altitudes under the assumptions that there is a
break from a=l.6 to 2.1 at 3Xl0 14 eV in the
differential spectrum of primary nucleons and
'=0.5 and l=80 g cm- 2 •
1----1----4-4---~-+-+-\-1
Fig. 10. Integral spectrum of nucleons
at various altitudes under the same
assumptions as in Fig. 9.
High Energy Gamma-Ray s in the Atmosphere
119
smoothed out, as nucleons come down through the atmosphere in such a
way that the spectrum gradually changes its slope. Then the bending
point is defined as the energy, at which the power index of the integral
spectrum is equal to (a1 + a 2 ) /2. Referring to the altitude variations of
the differential and integral spectra shown respectively in Figs. 9 and 10,
we are able to obtain the shift of the bending point
(3·3)
This is shown both in Figs. 9 and 10 for the values of the parameters
given in (3 ·1) and (3 · 2). The dependence of the bending point on the
elasticity coefficient is given in Fig. 11. The shift is maximum for (=0.575
but depends rather weakly on the value of (. For ( around 1/2, the 1/e
fold length of the bending point is approximate ly given by
(3·4)
This indicates that the shape of the N-componen t spectrum varies
rather slowly with atmospheric depth. At Mt. Norikura the value of Eb
is about 30 Te V, and therefore theN-compo nent spectrum shown in Fig. 6
may be regarded as representing the part above the bending point. In
fact, the spectral index given in (2 ·12) is in agreement with a2 but not
with a1.
I
13
Or---~~--~~-~----~~~
200
400
600
800
1000
(gcni 2 ) ....___
___,__ _~___,__
___.__ __....___
__.
Fig. 11. Depth dependence of the break point in the energy spectrum. Here the break
point is defined as the energy at which the slope of the integral spectrum is (a1 +a2)j2.
S. Hayakawa, J. Nishimura and Y. Yamamoto
120
§4. Spectrum and altitude variation of T-rays
The energy spectrum of the N-component is transferred to that of rrays. Referring to the mathematical detail given in Appendix A, we here
discuss the bending of the· r-ray spectrum, which is probably attributed to
that of the primary spectrum. In addition, the altitude dependence of the
r-ray intensity is analyzed. A part of the result is reproduced in II; here
we take account of the r-ray intensities at very high altitudes, in order to
demonstrate that our choice of the attenuation length can also account for
these data and to facilitate the analysis of primary r-rays, as employed in
our different paper. 39)
If the integral production spectrum of neutral pions produced between
x and x dx is expressed by a single power function as
(4·1)
the integral spectrum of r-rays and electrons at x IS given, according to
(A·8), as
(4·2)
where L is the attenuation length of nuclear active particles and P(/3, x, L)
is given in (A· 9). Since L is larger than the radiation length Ao, the
altitude dependence at small .x is represented by
P({3,x,L)=="
Nl(/3)
{;.1(/3) +_L}x+
M(/3)
{;.2(/3) +_L~x.
1 + 'h (/3) L I Ao
Ao
L
1 + A2 (/3) L I Ao
Ao
L J
(4·3)
As x Increases, the term proportional to exp(
over others, so that
x/ L) becomes dominant
P(/3
L)----{
N1(/3)
M(/3)
} -x!L
'x, - 1+J.-;(t3)LIJ.o- 1+J2(/3)LIJo e ·
(4·4)
Thus the E-component intensity increases linearly at small depths, passes
through a maximum and then decreases exponentially, as x increases,
whereas the energy spectrum of r-rays does not change at all, so long as
{3 is constant.
Taking these qualitative features into account, we are able to determine
the values of {3 and L by comparing (4·2) with observed data. For E'¥>1
Te V, the choice of
{3=2.3
and
L= 100-110 g cm- 2
results in an overall fit, as shown in Fig. 2.
As energy decreases, the value of {3 gradually decreases.
(4·5)
This affects
High Energy
Gamma~Rays
in the Atmosphere
121
the altitude variation only at small depths; for example, the depth at
which the intensity is maximum is shifted towards greater x, as ~ decreases.
However, the experimental data now available are not enough to distinguish
such fine structures of the altitude variations. We therefore limit ourselves
to discuss the altitude variation of the bending point of the energy spectrum
in a qualitative way.
The relation between the bending point of the observed spectrum of
r-rays and the production spectrum of neutral pions are somewhat com~
plicated. Firstly, the bending point shifts to the low energy side by the
decay of neutral pions. Secondary, observed r~rays have experienced the
atmospheric cascade starting from the parent r-rays, whereby the bending
point is also shifted to the low energy side. The amount of the shift, of
course, depends on the spectral index ~ as well as the attenuation length
L, as discussed in Appendix C.
Here we summarize the results only. Assuming (/31 + ~ 2 ) /2 = 2, L = 110
g cm- 2, the bending point of the E-component behaves as
(4·6)
where Eb(O)e-xiLb is the break point of the nucleon spectrum, g the ratio
of the average energy of pions to the energy of a parent nuclear active
particle. The factor 0.40 comes from the degradation due to the neutral
pion decay and shower development, in which the contributions of respective factors are about the same.
Since g is roughly several per cent, the bending of the E-component
spectrum can be reasonably attributed to that of the primary spectrum.
The bending point of the spectrum of the nuclear active component at
several tens of Te V corresponds to the bending point of about 1 Te V
for r-rays. This seems to be the case, as seen in the r-ray spectrum at
Mt. Norikura.
§5. Comparison between the r-ray spectrum
and the muon intensity
The intensities of r-rays and muons are directly related if they arise
only from pions. Actually, however, kaons contribute appreciably to
muons but only slightly to r-rays. Hence the comparison between them
would tell us the K/n: ratio for the primary energy up to several hundreds
of TeV.
The integral energy spectrum of r-rays at depth x is given, according
to (A·33), as
Je,y(E,x)
___
P_o..:.(~-'-'x___,, L) _____ J/J. (E),
B7J"(E, L)r;7J"(~, r) (~+ 1)
(5·1)
122
S. Hayakawa, J. Nishimura and Y. Yamamoto
where J!J.(E) is the vertical flux of muons at sea level and
(5·2)
(5·3)
with
(5·4)
The notations L, l and fJ are the same as those in the preceding sections
and -r1r is the mean lifetime of the charged pion and h0:=::::6.4 X 105 em is
the scale height of the isothermal atmosphere.
Adopting J!J.(E) given in Fig. 5, we obtain the r-ray spectrum expected
at Mt. Norikura; this is given by a curve indicated by JK 0 in Fig. 12.
This is appreciably larger than the observed intensity, even if the uncer~
tainty in the absolute fluxes of r-rays and muons is taken into account.
This may be accounted for by the contribution of kaons.
The contribution of kaons is represented by a parameter
(5·5)
the ratio of the integral production spectra corresponding to muon energy
E. The intensity of muons arising from kaons is calculated in detail in
Appendix B.
Kaons contribute also to r-rays mainly through the decay of neutral
kaons. This gives a correction factor Q(E,JK)
(5·6)
where the quantities in (5·6) are defined in Appendix B.
factor (5 ·1) is corrected for as
P(fJ, x, L)J!J.(E)
B1r(E)r;1r(fj) (fJ+ 1)Q(E,jK) .
With this
(5·7)
'The r-ray spectra given by (5 ·1) are shown in Fig. 12 for several
values of JK. This comparison may give us the most probable value of
JK as
(5·8)
High Energy Gamma-Rays in the Atmosphere
123
/cm 2 sec sr
-I
10~---4----------+-~~-----r--~
1013
ev
Fig. 12. Integral r-ray spectrum expected at Mt. Norikura from the muon spectrum
given in Fig. 3 for a numbe<o£ values of the K/7!: ratio fK. Experimental values
are those given in II.
On account of the difficulties in determining the absolute fluxes and the
statistical accuracy, however, we may only say that JK is finite but not
greater than unity. Such inaccuracy is inherent to this kind of methods,
but this result is not inconsistent with the K/rr: ratio observed by a large
emulsion stack. 24>
§6. Neutrino flux
Neutrinos are produced in various decay processes and their flux is
related closely to the fluxes of muons and r-rays. However, there are
differences between muons and neutrinos in the following respects.
124
S. Hayakawa,
J. Nishimura and Y. Yamamoto
Firstly, the contribution of kaons to muon neutrinos is comparable to
or greater than that of pions, because the energy of a neutrino in the rest
system of the parent particle is much greater in the former case. Secondly,
the flux at a large zenith angle is considerably larger than the vertical
flux, because the path length is larger in the former case. This is also
the case for muons, but the horizontal to vertical ratio is much greater
for electron-neutrinos which arises from particles of longer lifetimes.
Thirdly, a considerable fraction of Pe comes from Kg rather than from p-.
An analytic method of deriving the muon flux is given in Appendix
A. Although this is reasonably accurate for n~p+v and K~p+v decays,
the contribution of muon decays has to be evaluated numerically. 40 ) On
account of the importance of horizontal muons, furthermore, the numerical
calculation taking account of the spherical atmosphere is welcome. 41) Here
/cm 2 sec sr
-2
10
~---r----~---+----+----+----+---~
Fig. 13a. Vertical fluxes of neutrinos originated in the atmosphere. The full curves
show the muon-neutrinos (vp.) and the dashed curves the electron-neutrinos (ve).
The curves attached with K represent the neutrinos originated in kaons under
the assumption of K/n: ratio to be 0.2 over all energy region. Space neutrino
fluxes are also indicated.
High Energy Gamma-Rays in the Atmosphere
125
/cm2 sec sr
Fig. 13b.
Horizontal fluxes of neutrinos analogous to those in Fig. 13a.
we refer to the result of the latter.
In this calculation it is assumed that the production spectra of pions
and kaons are of a power shape with [3= 1.67, the K/n: production ratio
is 20%, and the K+ / K- ratio is very large. The production spectrum
adopted may not be far from ours below 1 TeV; in this energy region we
are mainly interested, because the flux of neutrinos with energy higher
than this region is too low to detect. 'The results thus obtained are shown
in Figs. 13a and b.
It may be interesting to note the feasibility of detecting neutrino interactions with this flux. Let us consider the neutrinos of energies greater
than 10 Ge V, because the neutrino flux of such high energies available with
existing accelerators is too weak. The intensity of all kinds of such neutrinos is of the order of 10-3 cm- 2 sec- 1 sr- 1 • If the interaction cross section
is about 10-38 cm2 , the rate of interactions in an omnidirectional detector
is estimated as about 10-3 ton- 1 month- 1 • Thus we need a detector of the
size as large as (10m) 3 for performing such experiment. If we do not
S. Hayakawa, J. Nishimura and Y. Yamamoto
126
detect interactions directly but observe muons produced, the range of muons
is equal to the thickness of matter above the detector. If we use a detector
of 100m2 sr, the counting rate of such muons is estimated to be of the
order of one count per month. If the cross section increases with energy,
the rate of interactions would be higher by two orders of magnitude.
Before concluding this paper, we briefly, mention the flux of cosmic
neutrinos. These neutrinos are produced by nuclear interactions with
matter in space. Since cosmic r-rays are also produced by the same
processes, the neutrino intensity can be estimated in the same way as the
r-ray intensity. 39) The intensities of respective components of neutrinos
thus estimated are shown in Figs. 13a and b. They are far smaller than
the intensities of atmospheric neutrinos except at very high energies.
Appendix A
Analytic methods for deriving the r-ray spectrum
from the muon spectrum
Altitude variation of the r-ray intensity
If the intensity of incident electrons and r-rays with energies between
E and E dE respectively are represented by simple power forms;
a.
13
rr:(E, O)dE
Eo) dE
K, ( E
'
r(E, O)dE
Eo \ 13 dE
K 1 (E)
,
(A·1)
the differential spectra of electrons and r-rays at a depth x are given by
linear combinations of the two well known elementary solutions42 > as
Here ,{ 0 37.1 g cm- 2 is the radiation length in air, and a1 and a 2 are constants which are chosen to satisfy the initial conditions
1,
a1
O'o
C({3)
+a2
+·..<1 ({3)
O'o
C([3)
-1
+ A2 ([3) - '
(A·3)
where 0'0 , C({3) and . < 1, 2({3) are familiar parameters in cascade theory. 42 >
The production spectrum of r-rays, g1 (E 1 )dE1 is connected with that
of pions g7To(E7T)dE'Tr as
(A·4)
High Energy Gamma-Rays in the Atmosphere
127
where P1r is the momentum of a pion produced and can be put equal to
E1r in the extremely relativistic case concerned. The last expression is
obtained by assuming the power spectrum of spectral index {j, and g1r(E1r)
= 2g7T"o(E7T") is the production spectrum of charged pions.
Let the production rate of parent r-rays as the decay products of
neutral pions with energies between E and E +dE and at depths between
x and x+dx be
Ur(E,x)dEdx=([j+l)- 1 g~( IJl)se-xiL~
d{,
(A·5)
where a single power law of the production spectrum and an exponential
decline of the source intensity with attenuation length L is assumed for
simplicity. The unidirectional intensity of electrons generated by those
parent r- rays is given
je(E, x) =L-1 (S+ 1)- 1 u~( IJl
(S + 1) -1
)s ~ ~:n(E, x-x')e-x'/Ldx'
u~(-1!) s ~ [ a~~1A::x:~~T11~;: + a2 ;A~~;:~~){;~: J.
(A·6)
The intensity of r-rays is given by replacing a1 and
respectively by k1 (S) and k2 (S), which are expressed as
a2
in (A· 6)
(A·7)
The values of
a1, a2, k1
and
k2
are given by
The integral intensity of the electronic component with energies greater
than E is obtained as
J.,r(E, x)
~: {j.(E', x) + jr(E', x)} dE'
1
S+ G7T"(E)P(S, x, L),
1
(A·8)
where
P(S, x, L)
(A·9)
with
(A·10)
and L - 1 G1r(E) is the integral production spectrum of charged pions. The
numerical values of N1([j) and M(S) are given in Table A·l.
S. Hayakawa, J. Nishimura and Y. Yamamoto
128
Table A·l. Numerical values of N1(f3) and Nz(f3).
3.0
1.226
-0.226
b.
Muon flux at sea level
Hereafter we specify the zenith angle fJ and define x as the atmospheric depth measured in the vertical direction. Hence x in (A· 8) and
(A·9) should be replaced by x/cosfJ.
Charged pions produced are lost by nuclear collisions with the mean
free path l as well as by spontaneous decays with the mean flight path
ha,
(A ·11)
where 'C'1r is the mean lifetime and m1r the mass of the pwn.
differential flux of pions is given by
Hence the
where ho = 6.4 X 105 em is the scale height of the isothermal atmosphere.
Putting
(A·13)
we express (A ·12) as
j1r(E1r, x, fJ) =g1r(E1r)e-x!tcos()
se~
A1r( (+- l) c:SfJ
, se), (A·14)
where
(A·15)
Multiplying f1r by the decay probability se/ x times the energy spectrum
of muons m1rdEfJ.j2p!p7r and integrating it over the energy range of parent
pions between E; and E;., we obtain the differential production spectrum
of muons as
gp. (E p., x, (})
=
X
1
X
)
L -1 e-x/Lcose\E;
JE;g7r (E 7r ) A 7r ((-l-L1 ) cos
fJ ' se
_'f!!.:::_dE7r
2p!p7r
'
(A·16)
where p! is the momentum of a decay muon 1n the rest system of the
High Energy Gamma-Rays in the Atmosphere
129
pion. The differential intensity of muons with the zenith angle () at the
depth x is obtained as
f,.(E,., x, 0) =
~:g,.(E,., x', O)dx',
(A·17)
provided that the decay and the energy loss of muons are neglected.
A remark must be made on the validity of (A·17) at large 0. At ()
close to rc/2, we have to take the curvature of the earth's surface into
account. Because of this effect, the zenith angle at an observing level is
different from that at a production level. The latter, O(x), is given by
sin2 ()
cosO(x) = [ 1- ( 1 + h(x) I R) 2
]1/2
(A·18)
,
where R is the radius of the earth and h(x) is the height of the production level,
h (X)
=
ho ln ( Xo/ X),
being the atmospheric depth of the observing· level. At large (), () in
(A ·12) ..._.(A ·17) should be replaced by O(x) .43)
The next task is to evaluate (A ·17) with (A ·16). Since the integrand
in (A ·16) involves an incomplete r-function, the integral has to be
evaluated numerically. In order to avoid such complexities, the integral
in (A ·16) is evaluated under the following approximations.
Xo
Approximation 1.36 ) On account of that the range of the integral in
narrow, the mean value theorem is applied. Using the mean values
(A·19)
we have from (A ·16)
f,.(E,., x, 0)
=
8g7r (aE,.) · B(x, sf}),
(A·20)
where 8=m7r/m"", rn,. is the muon mass and
x'
cosO
,
59 )e-x'/Lcos£Jdx'
exp [--,- (x/ l cos 0) (1- t (lj L)t)] {sodt.
l
--se
cosO ~ 1
1
L
o
1-t+ (l/L)t
(A·21)
The integral in (A· 21) is evaluated by neglecting the exponential term in
the integrand for x>I cos() and by expanding the integrand In a power
series as
for x"}>l cos(),
(A·22)
S. Hayakawa, J. Nishimura andY. Yamamoto
130
where a=l-(l/L).
Asymptotic forms of B1r(se) are given by
B1r (se) ~se cos 0I (se
1)
B1f(se)~se cosO [l/ (L -l)]
ln(L/l)
for se>1,
(A·23a)
for se~l
(A·23b)
and in intermediate region is given by
(A·23c)
Since B(x, se) is practically independent of x for x>L, the integral
1n (A ·17) is readily evaluated, and we finally obtain
(A·24)
Approximation II.*) The expression (A ·17) involves double integrals.
Now we carry out the integral over x first and obtain the differential
spectrum of muons at sea level as
(A·25)
neglecting the x-dependence of A as before.
Now A can be replaced by
(A·26)
where
dlnB(se) ~[ 1 + selln(L/l)J- 1 •
dlnE
L-l
(A· 27)
is .a parameter assumed to be independent of energy for a moment. Then
the integral in (A· 25) can be easily evaluated and we obtain
(A·28)
The difference between (A· 24) and (A· 28) lies in factors which are
functions of m 1j m1r and energy E11-, but the ratio of these two is weakly
energy dependent and given by
(A· 29)
'I"'he numerical values of this ratio are shown in Fig. A ·1 for l = 80,
L = 110 g cm- 2•
The result is much simplified, if l = L. Then we have an exact
form
*)
This method is adopted in reference e) in Table I.
High Energy
1.0
Gamma~Rays
in the Atmosphere
131
10
l
. fu. Jll~ (Jl)
-----
0.9
II-
0.8
f3 = 1.7
r
-r-._
2.0
2.~
26
0.7
Fig. A·l. Energy dependence of the ratio .ftt(I)/f.u(II) given in (A·29) for /3=1.7,
2.0, 2.3 and 2.6.
j~~. (E~~., X, fJ) =COS f) [ 1- e-x/Lcos9] \E~ g7T (E7r)
.
JE;r
=
S9
S9+
1
cosfJ [1
(A·30)
where a= m1r cho/r7r::::::=116 Be V. If the spectral index {3 is a rational number,
the integration in (A· 30) can be performed analytically. For a simple
case of {3 2.0, we have
J~~.(E~~., fJ)
c~sfJ
-( E~~.
a
[cl( ~
r
+
( ~ ) + ~3
)31n ((E~~./a)+1
Ju
E ~~./a) + n2
(E)
7T
fJ.
'
(A·31)
where c;=1-o- 2; (i=1, 2, 3). The ratios of the approximate intensities to
the exact one, /~~.(I)//~~. (exact) and /~~.(II)//~~. (exact), are also shown in
Fig. A·2 for {3=2.0 and l=L. According to Fig. A·2 the intensity with
approximation II is satisfactory over all energy region and asymptotically
close to the exact one.
I
f,. (ll)/ t,.(exact)
1.0
0.9
10
IOZ
10 3 Bev
I
::::::::::.!i:.(~t)
0.8
Fig. A·2. The ratio fil)/f.u(exact) and f.u(II)/f.u(exaxt) for /3=2.0 and l=L.
S. Hayakawa, J. Nishimura andY. Yamamoto
132
c.
Relation between the r-ray flux and the muon flux
Integrating (A· 28) over energy, we have· the integral muon flux at
sea level as
J
{3
[ 1- B-2(13+re+l)
({3 + ro) ([3+ ro+ l)
cos(} Brr (so) Grr (EP.).
1 _ 8_2
(A· 32)
JP.(Ep., 0)
Comparing (A· 32) with (A· 8), we obtain the relation between the r-ray
flux at the depth x and the muon flux at sea level;
J.
e,r
(E
)
,x
JcosOBrr(so)··~~P({3, x, L) l (E )
,fJ.
2
([3+ro) (S+ro+ 1)[
1 B[3({3+1)
1 B-2c1Hro+l)
=
(A·33)
Appendix B
Contribution of kaons
The lifetimes and the branching ratios of the decays of the charged
and neutral kaons are summarized in Table B ·1.
Table B ·1.
Lifetimes and branching ratios of the decays of kaons
Lifetime
K± (1.227 ±0.008) X 10.:.8 sec
K~ (0.90±0.02) X 10-10 sec
Decay mode
,u.±+v
,..±+n-o
,..± +n-± +n-+
e± +v+n- 0
,u.± +v+n-o
,...±+,...o+,...o
I!
?&'++n-"
,..o +n-o
e+v+,...+
.u.+v+n-+
Kg (6.2:tnxw-ssec
Branching ratio (%)
e± +v+,...+
,u.±+v+,...+
n:+ +n--+n-o
n:o+,...o+,...o
21&'
64.2±1.3
18.6±0. 9
5. 7±0.3
5.0±0.5
4.8±0.6
1.7±0.2
69.4+1.3
30.6±1.0
}
0.1
28. 3±5. 9
25.0±5.9
8.7±2.3
38 ±7
----0.1
First, the decay of Kg is neglected, because its lifetime is considerably
longer than the pion lifetime. Secondly, we take only such decay modes
of large branching ratios that give major contributions to muons as well
as r-rays. They are
K±_,. fl.±+ v (2/3),
(B·l)
High Energy Gamma-Rays in the Atmosphere
133
K~~n++n-
(2/3),
(B·2)
K~~n°+n°
(1/3),
(B·3)
where the branching ratios adopted are . shown in parentheses. The
contributions of other modes are so small that they are estimated as a
correction to the contributions of the major modes.
The process (B·l) is similar to the n-p, decay discussed in Appendix
A. The integral energy spectrum of muons from (B ·1) is thus given,
corresponding to (A· 32), as
(B·4)
where suffix K indicates quantities concerning the kaon, and GK'1::. is the
integral production spectrum of both kinds of kaons K+ and K-. DK represents the K-p, mass ratio
In the processes (B · 2) and (B · 3), the differential prodU<::tion spectrum
of pions is obtained from that of kaons as
(B·5)
where
(B·6)
The muon flux arising from the process (B · 2) is therefore, given by
in deriving (B · 7) it is implicitly assumed that the time delay in the K~
decay is negligible (BK~(E) ~1).
The process (B· 3) contributes to the electronic component. Its flux
IS evaluated in a way analogous to (A· 5) as
(B·B)
where P(fj, x, L) is given by. (A· 9).
Thirdly, we consider the correction for above choice of branching
134
S. Hayakawa, J. Nishimura and Y. Yamamoto
ratios of kaons by the treatment, in which the energy spectrum of decay
products is evaluated by the mean value theorem, that is, by the approximation I in Appendix A. Namely, in the K~~<2 and K1r 2 modes the energies
of the muon and the pion from kaons of energy EK are approximately
given respectively by
1
Efi.=TEK
for K....-)>p.+v,
E1r=_l_EK
2
for K....-)>2rc,
(B·9)
(B·10)
this is compared with Efl. (mJ1./m1r)E7r for rc....-)>p.+v.
Let us consider the decay of the i-th particle, whose energy spectrum
is g;(E;i)ocEij<!3+l)dEu, into the j-th particle with the decay probability
Bt(E;j) shown in Fig. B·l, the branching ratio P;i and the number of
the j-th particles n;h where Eii is determined by (B · 5), (B · 6) and similar
ones. The energy spectrum of muons or r-rays is given by summing up
all possible modes producing them as
(B·ll)
If muons or r-rays are produced by two-step decays, their energy
spectrum is given by
(B·l2)
The two-step process is of practical importance only for the K-rc-p. decay,
because the K-Tr;-r decay can be accounted for by the one-step process. If
we take the all decay modes in Table B ·1 into account, the flux of the
muons given by the summation of (B · 4) and (B · 7) should be corrected
Fig. B·l. Energy dependence of the decay probability of n-*, K*,
with 1=80 g cm-2 and L=llO g cm-2 •
K~
and Kg
High Energy Gamma-Rays in the Atmosphere
135
by a multiplicative factor ep.(E) and that of the r-rays given by (B · 8)
by a factor er(E). These correction factors are given by
The numerical values of these factors are shown in Fig. B · 2 assuming
Hence the
p,-r
RK (E)
ratio is given by
{f~l) (E)
+/J2) (E)} I f1l) (E).
(B·l4)
Here again are assumed the power production spectrum of E-a, and for
the equal production rates of four kinds of kaons, (B ·14) is reduced to
BK±(2E)of3+ BK~(2oE)B7r(oE) ep.(E)
BK~(4E)"("o/2)f3
ey(Ilf
(B·l5)
The value of ep.(E)/er(E) is also shown in Fig. B·2.
1.8
1\
1.6
~e,(£J
1.4
1.2
1.0
-...
Fig. B · 2.
~
102 Bel/
/
0.8
0.6
~
"~
I ~
~c.,tevcfl
.....-c'
Energy dependence of the correction factors 8 7 ( E) and
branching ratios of kaon decays shown in (B·l-..3).
c,l E) for
the
'This is compared with the p,-r ratio R1r(E) in the case that pions
are only a source of those secondary particles,
(f)
13
R1r(E) =B1r±(oE)
•
(B·16)
The values of B; (E) are given in Fig. B ·1. From this we see that
is nearly equal to unity below 104 BeV, because the lifetime of
K~ is very short. Then we have
BK~(E)
S. Hayakawa, }. Nishimura and Y. Yamamoto
136
BK±(2E)CJS+ B'lf±(CJE) e!J.(E)
---e"'(E)...
B'lf±(CJE)
(B·17)
The numeri cal value of the ratio RK/ R'lf is easily obtaine d by referen ce
to the Figs. B · 1 and B · 2. This value varies from 2 to 9 as the energy
increas es from 0.1 Te V to 10 Te V.
Appendix C
Bendin g points in the energy spectra of nucleons and r-rays
Bendin g points in the nucleon spectru m (I)
We choose the primary nucleon spectru m as consisti ng of two parts
which have differen t power shapes:
a)
fn(E, 0) =/~ )(E, O) +j~
1
f~t>(E, O)dE= ~~~'at( ~c
2
'(E, 0),
)al dE
j~ 2 '(E, O)dE=f~~>a2( ~c)a
2
c;;
(C·1a) .
Ed::_Ec.
for
(C·1b)
These two are connec ted at the point E = Ec as
.{'(1)
no a1
J
.{'(2)
nO a2
.}
(C·2)
•
Since the contrib ution of charged pions to theN-c ompon ent is regarde d
as being of minor importa nce, we take only nucleon s into accoun t for
our illustra tive analysis .
When a nucleon penetra tes through the thickne ss x, the probab ility
for the nucleon to make r collisio ns is
P,(x)
(x/l)r e-x/1
-··---
r!
(C·3)
'
where l represe nts the collisio n mean free path of nucleon s. By a nuclear
collisio n an inciden t nucleon of energy E is assume d to survive with
energy (E, and after r collisions, therefo re the energy of the nucleon is
reduced by a factor of ('. We conside r for simplic ity C to be constan t
1n all energy region. Hence the energy spectru m at x is given by
fn(E, x)
£ {j~ncc-rE, 0) +/~2 '(c-rE, O)} P,(x),
r=O
~
=j.P>(E,
2
O)~Cra 1 P,(x) +f~ >(E,
r=O
~
0)
~
(ra 2P,(x).
(C·4)
r<=Yc+l
The critical numbe r rc is an integer determ ined on accoun t of (C·1) by
High Energy Gamma-Rays in the Atmosphere
137
(C·S)
For E2Ec, only the second term in the last expression of (C · 4) is
retained, because only higher part of the primary spectrum contributes,
and we have
00
fn(E,x)=j~ 2 >(E,O)~t;'a 2 P,(x)=j~ 2 >(E,O)e-xiLz
r=O
for Ed;:.Ec,
(C·6)
where
(C·7a)
is the attenuation length of the nucleons in the energy region EL_E,.
The attenuation length in the low energy region, E<E,, is analogously
defined as
(C·7b)
For E<E" two parts of the primary spectrum contribute, so that
rc
fn(E, x) = j~ 2 >(E, O)e-x1L 2 + ~ [f~n(E, O)t;'a 1 _
r=O
/~ 2 > (E,
O)t;'a 2 ]p,.(x)
(x/ l)' -~a x/IJ
(_ji__)az-al}
E,t;'
r! e
2
for E<E,.
(C·8)
In practice we deal with the integral spectrum rather than the differential spectrum. The primary integral spectrum is given by
J:i'( ~
F.(E, 0)-
r
for E';?:E, (
=J:~'( ~· )"'[1 (1- ::-)(;,)"']
(C·9)
for E<E,.
f
In (C · 9) two parts are connected rather smoothly, as shown in Fig. 11.
The integral nucleon spectrum at the depth x is given by carrying out of
the integration in (C · 6) and (C · 8) over energy as
Fn(E, x) =FJ2>(E, O)e-•1' 2
for E?::_Ec,
Fn(E, x). F~2) (E, O)e-x/Lz[1- ~
r=O
{1-
a2
(C·10)
(~)a2-al} (x/l)" e_(;a2x/IJ
al
for E<Ec,
'Ec
r!
(C·11)
where
(C·12)
S. Hayakawa, ]. Nishimura and Y. Yamamoto
138
is the integral spectrum of primary nucleons in the region E2Ec. Now
we define the bending point as the energy at which the spectral index
equals to Ca1 + a 2) /2. This is obtained to be
(C·13)
which is smaller than the break point Ec.
The same procedure is applied to obtain the bending point of the
integral spectrum at any depth, and this shifts with increasing depth as
(C ·14)
The
1/e
fold length of the bending point 1s
(C ·15)
Since Lb~5l for a 1 1.6, a 2 2.1 ·and
0.5, the shape of the N-component
spectrum varies rather slowly with depth.
Bending points in the nucleon spectrum (II)
In obtaining the bending point in the r-ray spectrum, a more sophisticated method may be useful. With the same notations as in the preceding
subsection a), the energy spectrum of primary particles is expressed by
(C ·1) assuming a bend at energy Ec. The altitude variation of the energy
spectrum of the nuclear active component is given, with the help of the
Mellin transformation, by
b)
\d ( Ec )s dE
f n (E ' X ) =.{(2)~
27f'i j s E
e
-(1-(;•)x/1
nO
(C·16)
The integral spectrum at a depth x is
(C·17)
The same results as (C·10) and (C·11) are obtained from the above
formula by performing the integration after the expansiOn
The integration with respect to s is performed by the saddle point methods,
1n which the saddle point is given by the equation
Ec- - -1+ - C ln( x--~+----=0.
1
1
ln
E
s
l
s · a1 a2 s
(C ·18)
High Energy Gamma-Rays in the Atmosphere
139
The value of s thus obtained represents the spectral index and varies with
energy. Thus we obtain
8
_ .{'(2)
a2
( EE"'c ) eq(s),
F n (E 'X ) -JnO
V2rcq"(s)
where
(C ·19)
s is a saddle point and
q(s)
Ins- (1
q"(s) = -
~+~
(.S)x/l
1
1
ln(-- + --),
s-at
1
('(In()'+ (s a,)'
l
a2-s
+
1
(C·20)
(a, s)' · f
It may be appropriate to define the bending point as the energy at
which
(C · 21)
Substituting (C·21) in (C·18), we have the bending point as a function
of depth x:
(C·22)
The last factor represents an effect that the bending point in the integral
spectrum is shifted from that in the differential spectrum; that is
(C·23)
This is essentially equal to that given in (C·13). The first term in the
bracket in (C · 22) represents the shift due to the attenuation. The efolding length for this is given by
(C·24)
The expressiOn (C · 24) is an asymptotic form of (C ·15) for a 2 a 1~1.
For (=0.5 and Ca1+a2)/2~2 as in §3, the bending energy is expressed
simply as
(C·25)
c)
Bending points in the r-ray spectrum
Let the production rate of the neutral pions be similar to (A· 4) but
consist of two parts of different spectral indices:
l
f
(C·26)
S. Hayakawa, J. Nishimura and Y. Yamamoto
140
The depth dependence, g1fu(x), is assumed to be the same as that of
the nucleon spectrum, and we· have
g1fo(E, x)dE=g~ ~- \ ds( E} )s!ilf e-{l-(I-6•)}x//(-·_1_ + _1__ ).
/32 s
s- f3t
E .
E
2rr:z J
(C·27)
The spectrum of the r-rays generated in the decays of neutral pions is
obtained by the above formula,
Ur(E, x)dE=g~~ \ ___!/!__( E1 )s_dlf
E
2m J s+ 1 E
e-ft-ct-6•)}xl/(_1_
s-- {31
+-~-).
132-s
(C·28)
The integral energy spectrum of the electronic component generated from
the source (C · 28) is given by
Je,y= ~: dx'~~dE' [ll(E', E, x') +r(E', E, x')] gy(E', x-x'), (C· 29)
where ll and r are the integral cascade functions of electrons and r-rays
respectively. Substituting (C · 28) into (C · 29), we get
where . 1 1 is the same as in Appendix A.
The ''bending point is obtained in an analogous way as
1
1
s
s+1
AoC ln(
l
-··--··-X
J~-JoCClnC)/l
..1t+.1o(f C)/l
(C· 31)
and
(C · 32)
In obtaining the numerical factor, 0.4, in ( C · 32), we assume Cf3t + {32) /2
and other parameters are taken to be the same as those in §3. The
factor 0.4 in (C · 32) comes from the energy shift by n°--2r decay and
the· energy degradation by the development of cascade showers. The
former gives a factor 0.7 and the later the factor 0.55. The integration
of (C· 30) with saddle point method results in
High Energy Gamma-Rays zn the Atmosphere
141
(C·33)
where
s is
q1 (s)
the saddle point determined by (C · 31) and
ln s(s+ 1)
(1- C:.S) J.ox/ l
In {(1
C)J.o/1 + ,h}
+ln(--1-+ 1
.
s- f3t
/32 s
(C · 34)
It may be worth remarking that the depth dependence of the bending
point in the r-ray spectrum is the same as that of the nuclear active
component. This is due to an implicit assumption that the production
mechanism of neutral pions in nuclear interactions is essentially independent of energy.
Appendix D
Effect of energy dependence of energy loss rate of muons
Energy loss of. muons
I)
Ionization and excitation loss
The rate of energy loss of a muon of mass m!J,, momentum p and
velocity c/3 by ionization, excitation and Cerenkov radiation per g cm-2 is
expressed as 44)
a)
(D·1)
where
E~=--­
P2
E+m!~)/2me
(D·2)
is the maximum transferable energy (in MeV) to an electron and E 1s
the total energy of the muon. The last term in the curly bracket, 0,
represents the density effect and 1s given by
0=4.606loglo(-L) -C for p>Dm!J,c.
m!J,c
Parameters A, B, C and D
cteristic to the medium. The
tabulated in Table D ·1 for Al
rock 18) (Z= 12.8, A 25.8, Po
14.79, Po 1.025 g cm-2) .
(D·3)
in (D · 1) and (D · 3) are constants charanumerical values of these parameters are
and H 20 as respective representatives of
3.02 g cm-2 ) and sea water 20 )
7.433,
S. Hayakawa, J. Nishimura andY. Yamamoto
142
Table D·l.
B
A
c
D
Al
0.0740
16.77
4.21
1000
H20
0.0853
18.35
3.47
100
Pair creation loss
The energy loss rate by direct pair creation has been calculated by
Hayakawa and Tomonaga5) and by Mando and Ronchi, 45) but these results
are not accurate enough for quantitative purposes. A more accurate
calculation has been performed by Murota, Ueda and Tanaka.21 ) This is
expressed as
b)
(D·4)
dt1 is the differential cross-section for emitting an electron pair with
energies
and·
E+
E-,
and is given by
- (3+v2)} + i+: (1
-ln(
1
v2) +
~2 { 1~x + (2
~x) ·x(3+v2)} Jdudv,
v2)
(D·5)
where N is Loschmidt's number and
with u
The factor L represents the screening effect by the atomic electrons
and is given by
L
ln(183Z-113 V1+x)
for
(D·6)
where a 1s a factor of the order of unity. Thus (D·4) Is reduced for
rock and water respectively at about 1000 Be V to
-(
~!)"=1.76x10-6 E
MeV/gcm-2 ,
(D·7a)
1.09x10-6 E MeV/gcm-2 •
(D·7b)
High Energy Gamma-Rays zn the Atmosphere
143
Bremsstrahlung loss
The energy loss rates by bremsstrahlung are given for rock and water
in the cases of no screening and screening respectively by
c)
1
E 2 ) -0.308 } MeV /g cm-2
dx ) bc > 2.06 X 10-7E { ln ( m~'-c
( ·_!]_E
for rock,
=
1.23 X I0- 7E {1n( m~c
2 ) -0.127} MeV/gcm-
(D·8a)
2
for sea water,
1n the case of no screening, and
(
~:
2
): )
=2.2X 10-6E MeV/g cm-2
for rock,
(D·8b)
6
1.3 X 10- E MeV /g cm-
2
for sea water.
1n the case of complete screening.
Energy loss by nuclear interactions
The inelastic collision of a muon with a nucleus is regarded as the
interaction between the virtual photon cloud surrounding the muon and
the nucleus. For muons of energies up to a few tens of GeV, the
nuclear interaction cross-section can be accounted for, according to an
underground experiment by the Osaka City University group, 46> in terms
of the energy independent photonuclear cross-section of dr (1. 4 + 0. 3) X
I0-28 cm2 per nucleon and the Williams-Weizsa cker formula. 'The pion
production in this energy region is interpreted essentially by the onefire- ball model with inelasticity of K ~ 1/2.47) If this holds, the energy
loss rate is given by
d)
(D·9)
No reliable· information is yet available at higher energies. If, however,
the fire-ball model such as in the nucleon-nucleon collision holds also in
the muon-nucleon collision, the expression (D · 9) may be extended to
higher energy. The energy loss rate thus obtained is smaller by one
order of magnitude than the bremsstrahlung loss rate. The former is
therefore negligible compared with the latter in the first approximation
and may be taken into account later as a correction.
II)
Effect of variation of energy loss rate on depth-intensity relation 23)
When the muon energy exceeds lOCO Ge V, the energy loss by pair
S. Hayakawa, J. Nishimura and Y. Yamamoto
144
creation and bremsstrahlung is of primary importance. The range fluctuations due to these catastrophic energy loss processes are treated in
detail in Appendix E. Here the effects of energy dependence of the
average energy loss rate on the intensity-depth relation are considered.
The total energy loss rate is expressed as
(_c{~)
dx
a, bE
total
cln(
E~ 2 )MeV/gcm-2 •
mJ.l.c
(D·lO)
The values of parameters a and c are given as
a
1.84,
c=0.076
a
1.75,
c
0.085
for rock,
for sea water.
}
(D·ll)
b includes the effects by pair creation, bremsstrahlung and nuclear interactions. Its value at 2000 Ge V is estimated as
(D·12)
for rock with the reservation of the inaccuracy in the pair creation loss
and of the uncertainty in the nuclear interaction loss. For water this is
given by
b 2.5 x 10-e MeV /g cm-2 •
(D·l3)
The expressions (D ·12) and (D ·13) imply further inaccuracy, because
the screening effect in pair creation and bremsstrahlung is not complete
around 1000 GeV. Hence the value of b varies with energy. If the value
of b changes by an amount Llb, the depth measured in muon radiation
length should be changed by an amount Jt as
Lit
Lib
t
(D·14)
The ionization and excitation loss E per radiation length should be correspondingly changed by an amount JEr as
(D·15)
The ionization loss given by the first and third terms of (D ·10) also has
an energy dependence. This results in the change of the critical energy
by AE2 • Hence the resultant change of the critical energy is
(D·16)
This affects the intensity of muons by an amount JI as
High Energy Gamma-Rays in the Atmosphere
145
(D·17)
where r is the power index of the range
spectrum. 'I'he energy dependences of
.Jb/b and .JE2/ E are shown in Fig. D ·1.
Appendix E
Range fluctuations
of high energy muons23 )
The range of muons is determined
mainly by the ionization energy loss. As
energy increases, however, the energy
Fig. D ·1. The energy dependences
loss rate due to direct pair creation and
of Jb/b and d€2/E.
bremsstrahlung competes with that due
to ionization, and the former is subject to considerable fluctuations, because
the fraction of energy lost at one collision is large. Since the effect of
. fluctuations is significant in deriving the energy spectrum from the intensitydepth curve, a mathematical method of dealing with the fluctuations is
given in what follows.
Let the differential energy spectrum of muons at depth t be j(E, t)
where t is measured in units of the muon radiation length. Then the
diffusion equation for the muon spectrum is given by
f}j
A ,'+
1 e BE '
(E·1)
where A' is an operator representing the effects of radiation, direct pair
creation and photonuclear reactions, and the last term represents the effect
of ionization loss with a constant loss rate e.
Equation (E ·1) is analogous to the diffusion equation in cascade
shower theory, so that the same procedure can be applied for solving
(E ·1). If the last term in (E ·1) is neglected, we have simply to apply
the Mellin transformation. The Mellin transform of the function j is
expressed as
IDl(s, t)
=
~~Esj(E, t)dE.
Then the integra-differential equation
which is readily soluble. If IDl(s, t)
by the inverse Mellin transformation
IS
IS
(E·2)
reduced to a differential equation,
obtained, the function j is given
j(E, t) =1-. \ dsE-s- 1 IDl(s, t).
27rl Jc
(E·3)
S. Hayakawa, J. Nishimura and Y. Yamamoto
146
In the presence of the ionization loss term, we use the power series
expansion, as in the case of the Approximation B of cascade theory. In
place of (E · 3) we now have
j(E, t) =-21. \ dsE-s-1£ (- Ee ·)n¢n(s, t),
nt
Jc
(E·4)
n=O
where if> 0 (s, t) = IDC(s, t). It is convenient to express the power senes by
an integral representation, so that
(E·5)
where
ill?: (s, n, t) = n!
ill?: (s, 0, 0) =
if>n (s, t),
E~,
(E·6)
which are distinguished from IDC(s, t) in the absence of ionization loss.
The normalization (E · 6) corresponds to the initial condition that one
muon with energy Eo exists at t=O, which is represented as
j(E, O)
o(Eo
E).
Then j(E, t) in (E · 5) represents the probability of finding a muon in
the energy region between E and E+dE at depth t. The probability of
finding a muon at any energy is obtained by integrating (E · 5) over all
energy as
\"'.
J(Eo, t) = Jo;(E,
t)dE=
1ni Jds
\ (Eo
-e- )sr(s)SJC(s, -s, t),
2
(E·7)
where
IDC(s, q, t) ==E~SJC(s, q, t).
The last expression in (E·7) is obtained by the residue at a pole q
If the muons have an energy spectrum at t 0,
-s.
j(Eo, 0) = }odEo/ E~+\
the intensity at depth t is given by
I(t)
=j0 ~J(E 0 , t)dEo/E~ 11
j 0 e- 13 r(f3)IDC(/3,
/3, t).
(E·8)
The last expression is obtained by performing a contour integral over s
by making use of a pole at s= /3.
In the last expression of (E · 8), the dependence on the material
through which muons traverse is implied only in the factor e- 13 • Comparing
the muon intensities under two different material layers of the same thickness, therefore, efie can obtain the spectral index f3 as
147
High Energy Gamma-Rays in the Atmosphere
(E·9)
Now we have only to obtain IDC by solving the differential equation
· [ :t +A(s+q)
Jsncs, q, t)
(s+q)qSR(s, q
1, t),
(E·10)
with the initial condition (E · 6). Here A(s+q) is the Mellin transformation
of operator A' and is expressed as
(E·11)
where ah ap and apN are the probabilities for the fractional energy loss of
v in dv by bremsstrahlung, direct pair creation and photonuclear reactions,
respectively.
The differential equation (E ·10) is reduced by applying the Laplace
transformation
L (s, q, p,) =
~~e-~ 1 ?.n(s, q, t) dt
(E·12)
to a difference equation
{p,+A(s+q)}..L'(s,q,p,)
(s+q)q...C(s,q
1,p,)+8q,o·
(E·13)
The difference equation is solved as
1:(
)-r r(s+q+1)r(q+1)
s, q, p. - m1.::! (p,+A(s))r(s 1)
rr {p,+A(s+i)}
tt A(s+q+i)
{p, A(s
i=l
m)}q .
(E·l4)
This yields
00
~ SRn (s'
n=O
q) e-A(s+n)t'
(E·l5)
where
(s+q+1)r(q+1) lim
A(s+q+n) A(s+n)
r(s+1)
m-ll>"" {A(s) -A(s+n)} {A(s+m) -A(s+n)}q
94.(s, q)
ll'
X
A(s+q+i) A(s+n)
A(s+i)-A(s+m)
i=l
(i~n)
(E·16)
Here II' means the omission of a factor with
n.
For our purpose we only need ?.nn(s, s), and we have from (E·l6)
r(l-s) .
r (1 +s5-l~ --'---~-----"---;-~~~--+----~-~
?.nn (s, - s)
X
ll'
;~1
(i~n)
A(i) -A(s+n)
A(s+ i)- A(s+ n) ·
(E·l7)
S. Hayakawa, J. Nishimura and Y. Yamamoto
148
If s is an integer, the product TI' is reduced to
$
TI'
A(i) A(s+n}_=
A(s+i) -A(s+n)
·i=l
(i""'n)
f!l {A(i) -A(s+n)}
A(n)
1
A(s+n)
It
TI' {A(m+i) -A(s+n)}
i=l
s
where TI' in the denominator means the omission of a factor of i = s + n- m.
i=l
s
In the limit of infinite m, the product TI' in the denominator cancels with
; .. 1
{A(s+m)
A(s+n)}s in (E·17) ·except a factor lim{A(s+m)
A(s+n)}.
m...;.n
An infinitesimal factor thereof cancels with the infinity arising from r(1
as
limr(1-s
o_,..O
o) {A(s+n)
A(s+n-o)}
( - 1)s-l
or(s)
0 8A(~j-n)
8s
s)
.
Thus (E ·17) is reduced for an integer s to
(E·18)
This can be computed by summing up the finite number of terms. The
value of ~(s, - s) for a non-integer s can be obtained from (E ·17) in
the usual manner in cascade theory, or most simply by interporating its
values for integers.
Introducing (E·18) in (E·8), we have the intensity-depth relation
The intensity-depth relation with fluctuations given in (E ·19) is compared with that without fluctuations. The latter is obtained by putting
do+dp
1n (E ·11).
(E· 20)
dpN=bo(v) /v
Namely,
A(s+q)
(E· 21)
b(s+q).
Substituting (E · 21) in (E ·19), we have
Io(t) = ·o(_!?_)l3~
r(S+n)
J e
n ... o r(S+1)r(n+1)
·e-bCI3+n)t=
.o__!__(_E-)13
S e
J
1
(ebt
1)13 .
(E· 22)
This coincides with the expression given by Heitler. 87>
In order to see the degree of fluctuations in the more intensity at
great depths, the numerical values of (Io- I)/ Io is evaluated by using
(E·19) and (E· 22).
High Energy Gamma-Ray s in the Atmosphere
149
The calculation is made by assuming the complete screening. crosssections for the bremsstrahlu ng and the direct pair creation processes and
neglecting the contribution of the photonuclea r process. Thus the value
of b here adopted~ is
The summation over n in (E ·19) is made up to the 15th term, but
it converges so rapidly that only a few terms give significant contribution s
at large t. The results for t= 0.6, 0.8, 1.0, 1.5 and 2.0 are shown in
Table E·l.
Table E·l. Degree of the fluctuations: (I-Io)/1.
t: the depth measured in units of muon radiation length.
f3: spectral index.
I(t) for t>0.6 are calculated with (E ·19), whereas those for t=0.01 and 0.1
are with (E · 27).
~
0.01
0.1
0.6
0.8
1.0
1.5
2.0
2
0.000
0. 0011
0.04
0.075
0.12
0.27
0.46
3
0.0024
0.027
0.23
0.38
0.55
1.26
2.21
0.007
0.070
0.52
0.89
1. 36
3.63
4
i
I
I
I
7.89
For small t we have to take many terms for evaluating the summation
in (E ·19). Thus we have to use a different method for t~l, namely the
moment method.
For this purpose the Laplace transform of j(E) is expanded as
L(A)
(E·23)
where
(E·24)
is the n-th moment of the energy spectrum.
The power series (E · 23) is rewritten as
L(J.)
=
oo
"'\:""1
N~
,N
11
N!
N
"'\:""1
~N
C (
..
l) .. <En)<E>N-ne->..<E>.
(E·25)
150
S. Hayakawa, ]. Nishimura and Y. Yamamoto
Noting that
J(E)]lnd.i
fexp[JE
J
2 .dna(E-(E))
m
dEn
'
we obtain the inverse Laplace transform of (E · 25):
j(E)
(E·26)
where
N
f(N, Eo, t) =--=-=-- ~NCn(
1)n (En)(E)N-n.
n=O
The intensity-depth relation is then given by
I(t)
~~01 ~j(E)dE
\
=\_!]Eo__ ~f(N Eo t)[_E_N-1~(fL<E))
J E6+1 N=l ' '
dEN- 1
00
dE
d
=N~ -. d(E\ dEo
(
)N-1. d(E\
dE
J
E=O
1
. E'{t1 f(N, Eo, t)
I
<E>=O •
(E· 27)
The first term represents the intensity-depth relation without fluctuations.
Since the solution of (E · 27) is derived by evaluating the deviation from
the solution without fluctuations, it is expected that the convergence of the
series is rapid only for small t, in contrast to the solution (E ·19).
The numerical calculation is made up to t=0.2, and the results are
given in Table E ·1. It may be remarked that the summation up to the
fourth term gives a reasonable accurate result. The numerical results
obtained from (E · 19) and (E · 27) are connected smoothly for respective
values of [j.
Appendix F
Neutrino flux
High energy neutrinos in the atmosphere are produced almost exclusively by the decays of pions, kaons and muons. The neutrinos from
muons have to be taken into account, because we are interested in neutrinos
with energies greater than 1 GeV. Moreover, we have to distinguish
High Energy Gamma-Rays in the Atmosphere
151
between two kinds of neutrinos, the one associated with the muon (vp..)
and the other with the electron (ve). They are produced by the following
decay processes as
+ JJp.. (Pp..),
f.J.±~e± + Ve (Pe) + Pp.. (vp..),
n±~ p.±
(F·1)
(F·2)
and those given in Table B·l.
The neutrinos produced by (F ·1) can be dealt with the same way as
the muons from n-p. decays. Hence we have only to replace E~ and E;;.
in (C·9) by
E~=oo,
E;;.
Ee+ (1/4) (1 a- 2)m;
Ev(1 a-2)
~
Ev
(F·3)
where Ev is the energy of a neutrino produced and ii=m1r/mp... The last
expression is obtained by taking account of the smallness of (1-iJ-2)m;/4
compared with E!.
Since E;- E;;. is very large in the present case, the approximation I
in Appendix A is inaccurate, and we apply the approximation II to evaluate
the integral in (A·21) with (F·3). Thus the differential energy spectrum
of neutrinos at a great depth, x~L, is obtained as
(F·4)
The integral energy spectrum is
(F·5)
The intensity of neutrinos relative to that of muons is given on account
of (A· 32) as
(F·6)
This is of the order of 1/10.
This ratio depends sensitively on the mass ratio 8. Therefore, the
v-p, ratio in the Kp.. 2 decay is very close to unity, because
Thus the contribution of kaons to neutrinos is comparable to or greater
than that of pions, on account of that the K/ n production ratio is as large
as 1/4. The contribution of other decay modes of the kaons is negligible
compared with those of the above two sources.
152
S. Hayakawa, J. Nishimura and Y. Yamamoto
The decay of muons give two kinds of neutrinos through (F · 2).
The mean flight path hd for muons is given by replacing the mass and
the lifetime in (A ·11) by those of the muon. Neglecting the ionization
loss of muons, we have the differential energy spectrum of neutrinos with
zenith angle 0 at depth x as
(F·7)
where Uv.cv") is the energy spectrum of electron(muon)-neutrinos from a
muon with energy E11- and p(x) is the air density at depth x. The double
integrals in (F · 7) are evaluated numerically.
References
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
23)
24)
S. Hayakawa, Soryushiron no Kenkyu, III, (Iwanami, 1961), cited as I.
S. Hayakawa, Prog. Theor. Phys. 3 (1948), 199.
Y. Miyazaki, Phys. Rev. 76 (1949), 1733.
R. Marshak and H. Bethe, Phys. Rev. 72 (1947), 506.
S. Hayakawa and S. Tomonaga, Prog. Theor. Phys. 4 (1949), 287, 496.
K. Greisen, Phys. Rev. 73 (1948), 521.
S. Hayakawa, S. Machida and M. Umezawa, Soryushiron Kenkyu (mimeographed circular
in Japanese) 1 (1949), No. 1, 74.
S. Hayakawa and ]. Nishimura, Prog. Theor. Phys. 4 (1949), 578.
Y. Fujimoto and S. Hayakawa, Prog. Theor. Phys. 5 (1950), 144.
B. Rossi, Rev. Mod. Phys. 21 (1949); 104.
S. Hayakawa and ]. Nishimura, ]. Sci. Res. Inst. 44 (1949), 47; Prog. Theor. Phys. 4
(1949), 232.
W. Heitler and L. Janossy, Proc. Phys. Soc. 62 (1949), 374, 669.
M. Akashi, Z. Watanabe, A. Misaki, I. Mito, Y. Oyama, S. Tokunaga, S. Dake, K. Yokoi,
S. Hasegawa, ]. Nishimura, K. Niu, T. Taira, T. Ogata, Y. Tsuneoka, A. Nishio, Y. Fujimoto and N. Ogita, Prog. Theor. Phys. Suppl. in this issue, cited as II.
Y. Fujimoto, S. Hasegawa, M. Kazuno, N. Ogita, ]. Nishimura and K. Niu, Proceedings
of the Moscow Conference on Cosmic Rays 1 (1960), 41.
M. Akashi, K. Shimizu, Z. Watanabe, T. Ogata, N. Ogita, A. Misaki, I. Mito, S. Oyama,
S. Tokunaga. M. Tamura, Y. Fujimoto, S. Hasegawa, ]. Nishimura, K. Niu and K. Y okoi,
]. Phys. Soc. Japan (1962) Suppl. A-III, 427.
J. Nishimura, J. Phys. Soc. Japan (1962), Suppl. A-III, 452.
]. C. Barton, Phil. Mag. 6 (1961), 1271.
S. Higashi, T. Kitamura, M. Oda, Y. Tanaka and Y. Watase, Nuovo Cim. 32 (1964), 1.
S. Miyake, V. S. Narasimham and P. V. Ramana Murthy, Nuovo Cim. 32 (1964), 1505.
P. T. Hayman and A. W. Wolfendale, Proc. Phys. Soc. 80 (1962), 710.
J. L. Osborne, A. W. Wolfendale and N. S. Palmer, Proc. Phys. Soc. 84 (1964), 911.
S. Higashi, T. Kitamura, S. Miyamoto, Y. Mishima, T. Takahashi and Y. Watase, Lecture
at the annual meeting of the Physical Society of Japan, October, 1964.
T. Murota, A. Ueda and H. Tanaka, Prog. Theor. Phys, 16 (1956), 482.
S. Miyake, V. S. Narasimham and P. V. Ramana Murthy, Nuovo Cim. 32 (1964), 1524.
]. Nishimura, Proceedings of the International Conference on Cosmic Rays, at Jaipur
4 (1963), 224; Handbuch der Physik (Springer-Verlag, Heidelberg, 1965), Bd., 46/2.
M. Koshiba, Proceedings of the International Conference on Cosmic Rays, at Jaipur, 5
(1963), 293.
High Energy Gamma- Rays in the Atmosph ere
153
25) Y. Kamiya, S. Sagisaka, H. Ueno, S. Kato and Y. Sekido, ]. Phys. Soc. Japan 17 (1962),
Suppl. A-III, 315.
26) J. Pine, R. ]. Dairsson and K. Greisen, Nuovo Cim. 14 (1959), 1181.
P. ]. Hayman, N. S. Palmer and A. W. Wolfendale, Proc. Roy. Soc. A275 (1963), 391.
27) D. D. Krasilnikov, Proceedings of the Internation al Conference on Cosmic Rays, at Jaipur;
4 (1963), 124.
29) V. C. Wilson, Phys. Rev. 53 (1938), 337.
30) ]. Clay and A. Von Gernert, Physica 6 (1939), 497.
31) P. H. Barrett, L. M. Bollinger, G. Cocconi, Y. Eisenberg and K. Greisen, Rev. Mod. Phys.
24, (1952), 133.
32) P. ]. Hayman, N. S. Palmer and A. W. Wolfendale, Proc. Phys. Soc. 80 (1962), 800.
33) S. Lal, A. Subramani an, S. C. Tonwar and R. H. Vatcha, Phys. Letters, 14 (1965), 332.
34) V. V. Guseva, N. A. Dobrotin, N. G. Zelevinskaya, K. A. Kotelnikov, A.M. Lebedev and
S. A. Slavatinsky, J. Phys. Soc. Japan 17 (1962), Suppl. A-III, 375.
35) S. Miyake, J. Phys. Soc. Japan 18 (1963), 1226.
36) G. Brooke, P. ]. Hayman, Y. Kamiya and A. W. Wolfendale, Nature 198 (1963), 1293;
Proc. Phys. Soc. 83 (1964), 853.
37) L. T. Baradzei, V. I. Rubstov, Y. A. Smorodin, M. V. Solovyev and B. V. Tolkashev, J,
Phys. Soc. Japan 17 (1961) Suppl. A-III, 433.
Proceednigs of the Internation al Conference on Cosmic Rays, at Jaipur 5 (1963), 283.
38) G. Clark, H. Bradt, M. Lapointe, V. Domingo, I. Escobar, K. Murakami , K. Suga, Y.
Toyoda and ]. Hersil, Proceedings of the Internation al Conference on Cosmic Rays, at
Jaipur 4 (1963), 65.
39) S. Hayakawa, H. Okuda, Y. Tanaka and Y. Yamamoto, Prog. Theor. Phys. Suppl. No. 30
(1964).
40) G. T. Zatsepin and V. A. Kuzmin, Soviet Physics, JETP 14 (1962), 1294.
41) R. Cowsik, Yash Pal, T. N. Rengarajan and S. N. Tandon, Proceedings of the Internation al
Conference on Cosmic Rays, at Jaipur 4 (1963), 211.
42) B. Rossi and K. Greisen, Rev. Mod. Phys. 13 (1941), 240.
43) K. Maeda, J. Geophysical Research 69 (1964), 1725.
44) R. M. Sternheim er, Phys. Rev. 93 (1954), 351; 103 (1956), 511.
45) M. Manda and L. Ronchi, Nuovo Cim, 9 (1952), 517.
46) Y. Watase, S. Higashi, T. Kitamura, Y. Mitani, S. Miyamoto, T. Oshio, H. Shibata and
K. Watanabe, ]. Phys. Soc. Japan, 17 (1962), Suppl. A-III, 362.
47) Y. Fujimoto and S. Hayakawa, Handbuch der Physik (Springer-Verlag, Heidelberg, 1965),
Bd. 46/2.