AVALANCHE PROCESSES IN COSMIC RAYS

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• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP80-00809A000600260313-5
• Release Decision: RIPPUB
• Original Classification: C
• Document Page Count: 8
• Document Creation Date: December 22, 2016
• Document Release Date: August 4, 2011
• Sequence Number: 313
• Publication Date: November 1, 1949
• Content Type: REPORT

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FORFI(;N IDnCIJMENTS OR RADIO BROADCASTS CD NO.
CLASSIFICATION CoNFIDIN'rIR1 r N 7" 1 T I A
CENTRAL INTELLIGENCE AGENCY REPORT
Scientific Cosmic rays
Book
Moscow
1948
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A!'&4CRE PROCCSES IR CCSNIC RAM
S. L. Beleu'kiy
Phys Inst imeni F. H. TLobedev
Acad Sci USSR
T.4Gh Cr COILS
foreword
Introduction
Chapter I. ..?ssic Processes
1. Radiat" onal Reta r?L tion of ffiec*,rona and Photcn
?air Production. Introduction of "t-units" 16
7_ TnnlsRtiron Y.neAAR AnA tha f!mm~t.nn Rffat?.t .^A
3. Role of Different Processes is 7ariova Rnar.,7
Regions 39
Chapter II: Cascade Theory for the Region of High Energies
4. Ilasic Equations of the Cascade Theory for the
Region of High Energies
Solution of the Basic Equations
6. Distribution Functions of Electrons and Photons with
Respect to Energy and Depth 51
CLASSIFICATION CONFIDEAriA1. W1'~" " ] u.
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Chapter III. Cascade Theory Taking into Account Ionization Losses
8. Basic Equations and Their Solution by Snyder's
Method
9. Solution in Variables 'lambda' and 'a'
10. Cascade Curve
11. Calculation of the Wucber of Charged Particles
as a Function of Depth
Chapter IV. Energy Spectrum of Avalanche Particles
12.
Energy "Equilibria." Spectra
79
13.
Spectrum of Delta-electrons and of Decay Electrons
83
14.
Discussion of Obtained Results and Comparison with
Results of Other Authors
91
15.
Energy Spectrum of Electrons at Varioue Depths
98
16.
Average Energy of (.`harge& Fartlclec
103
l7.
Energy Spectrum of Avalanche Photons
106
Chapter Avalanche Theory for Heavy Elemente
18. Snttin? Up the Problems. General Relations
111
19. Calculation of the Quantities t and t2
l14
20. Number of Particleo at the Maximum of the Cascade
Curve and the Position of the Maximum
123
Chapter
VI. Transitional Effects
21, Transitional Effects in Cosmic Rays
128
Chapter
VII. Scattering of Avalanche Particles
22. Kinetic Equations
136
r3?
Ylal --stri-tion v su v .. vim.
JIE
24.
(for bull Angles of Deflection)
Eonmaltiplication Scattering of Avalanche Particles 161
25.
Distributive Function of Avalanche Particles for
Large Angles of Deflection
172
26.
Influence of Scatter Upon the Fors( of the Cascade
27.
Curve
Spatial Distrib:?.tion of Particles due to Single
Scattering
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Chapter VIII. Secondary Shovers Generated by Mesons
index 2hl
29. The Meson Spectrum in the Region of High Energies 217
30. Great Pulses Generated by Masons 221
31. Great Pulses :nd Spin of the Meson 232
Bibliography. 238
28. Ionization Showers
sent this theory solved many knotty problems in cosmic rays, especially the problem
of the formation of cosmic ray-,showers. Althr'ugh the avalanche theory was applied
to a limited field of ?phenomena in cosmic rays, the theory seuw essentially to be
a tnique unified theory, of hosmic radiation. It explains the vigorous development
of the cosmic ray theory in the past 10 years, The work of Soviet pb iciats-has
been important to the development of the avalanche theory. Results Df t'Lis work
include the method of solving the basic equatiore of the theory,tti? establishment
The avalanche theory, that is, the passage through matter by high-energy
electrons and photons, originated in 1937. Even during its early stage of develop-
Foreword
of the "soft' character of the avalanche particles' spectrum, the basic results with
respect to the ve4tterina of n.':alanche particles, the theory of Auger showers, &id
+An anluhlr,n of ,row n1?.h~w .w.nhl nwa.
In 1941 a ensnary of the avalanche theory appeared in an article by Rossi and
Qreisen entatle.1 'Coamic Rays," now available in Russian translation. to spite of
the merits of this nuaawry, It is Incomplete and in part obeblete. The present
work is mainly devoted to the problehe worked cut in recent years. The basis for
this book is the work ,r t.;viet paysic.'.ete and theoreticians, among whom we must
first mention the Mork of L. D. ?anden and I, Ye,*Tamm. A consideratlo part of
this book has been written on the basis of the work of the author himself.
This book discusses the theory of the e].ertro etic interactions of high-
energy electrons and photons with matter. Szperie'ental data is drawn only to
illustrate the most impc:ta:t conclusion, nanny, the "soft" nature of the spectrum
and meson spin.
The author expresses his gratitude to I. Ye. Tang and L. D. Landau for their
ca+.w:vly ad-,!w n is .~tz uctia'?? A- -loo to S. X. T--nc7. and V. .I Ve! 1rr for
their discussion of many problemr.
Introduction
We shall not touch upon here the *.uclear processes in cosmic rave, although
these processes are of great interest for present. physics, The thsorctical
explanation of nuclear processes, to which the origin of the mesor. is related, the
interaction of protons and neutrons with matter in the region of very high energies,
'stare," -- at present all these meet with great difficulties in regard to principleu
(8ympoe ,.m an the Meson, edited by I. be. Tatum, OTTI, 1947).
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On the other hand, the study of the electomagnetic interaction of cosmic
radiation with matter is based uvon the oucceseive applications of quantum-
relativistic electrodyaae ion which has been successfully applied up to nmw to
various areas in physics. The extrapolation of the ordinary quantum-mechanical
live to the field of extremely '-igh energies is the Specific method used for the
problems of this book. The doubts that existed several years ago concerning the
possibility 8f such an extrapolation turned out to be unfounded.
The application of quantum electrodynamics to cosmic radiation has led to
very fruitful results and permitted one to tear down both the quantitative and the
quaJ.itative walls around the phenomena studied. Moreover, by relying on the quantum
f
t
d
h
y o
e S
u
mechanical theory, which hap, been worked out thoroughly enough during t
cosmic rays, one can hope to separate the field of tosmic phenomena which is
connected with specific, nuclear interactions from the field of phenomena which is
due to electromagnetic interactions. Such a separation seems possible and very
important for the construction of a theory of nuclear effects.
In the Investigation cf the electromagnetic action of cosmic radiation the
interaction of electrons and photons with matter is most essential and interesting
(henceforth in this book, by the term "electron" we shall mean loth electrons and
positrons).
Passing through matter, electrons and photons cf high energies take part in
tbe followin= processes: (1) radiational retardation (electrons); (2) processed
of pair-formation (photons); (3) ionization losses (electrons) (1) Compton effect
(photons); and (5) Rutherford scattering (electrons).
These processes, with the exception of the latter two, are discussed in Chapter
L or lice yr,weui. uvvn. li, '.;I& .field of ti.-1 y-. ..s.... -- 4--+-+ -1. 1a TlaveA
by the first two processes.Bhabha and Heitler(Proc. R^y. Soc. 139, 432, 1937),
and also Carlson and Oppenheimer (Pbye. Rev. 51, 220, 193 T), in 1937 showed for the
first time that thece processes should lead t,) the fcrastion of electron and photon
"showers." Having been stopped (retarded) inthe nuclear field, the electron creates
a photon of energy equal in order of magnitude to the energy of the first electron.
The high-energy photon can form with definite probability an electron-poaiaron pair.
Each component of the pair, being Aubjected to radiational retardation, r3dl.ttea
a photon and so on. After many repetitions of such processes we obtain, instead
of the initial electron, a great namber of photons and charged particles of both
signs. During all this, however, the energy of the original is being broken up;
therefore, the ranber of particles vth energy greater than a given one at first
increases up to a certain maximum and after that the energy quiclty falls to zero.
The behavior of the 3lectrons and photons during all this is naturally to be described.
b,- some integral equations, such as the so-called equations of the oaEca'e theory.
ti-__`-'~'~'"
In Rh"hha'a and Meitner?s works. and aibo Carleou'o and
In the work of the
heo
he
ons
...
..
basic equav.
r
i 21:; ^ 193t 1,
Soc. lE
Soviet theoretician Landau (S. Landau and. G. Ramer, Proc. Roy.
method
ce-Meli.ln tr,naf;a~nti.,^.
the equations of the cascade theory, by relying upon the Lapla
They permittall =z
n a Wilson clvsd nhnmbex,
to clarify qualitatively the "shower" of particles appearing i
ion, Along with the
Into the showerD
played by ionization
particles there begins to emerge another important role, that
is asymptAic terms
which terms were
for the processes of radiational retardation endPair- normatioa
high energies, are
emnlnseA in the above-mentioned works and which hold true for
;ht 61aments. Without
eery one cannot answer
b
en, accurate account of the indicated facto--A in the cascade t
view, and OydY Purely
d
sses into
,
Proce
can
-
t
Wig---
of the electrons and
oonsiaeration, we a c sin for the distribution function o
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CONFIDENTIAL
photons very complicated integro-differential equations; their solution by mnaens
of the Laplace-Mellin method encounters great difficulties.
In the course of the past 10 years, attempts have been undertaken to make
the cascade theory more accurate and precise. The first essential progress was
made by Snyder's works (Phys. Rev. 53, 960, 1938) and Berber's (Phys. Rev. 54,
317, 1938), in which they solved the equations of the cascade theory, taking into
onsideration ionization losses. As a result they obtained the so-called cascade
curve, that is, Vie full number of particles as a function of the depth of the
layer penetrated by the shower. Firstly, however, in the expression obtained by
Snyder and Berber there was introduced a complicated function, taken from an
equation in finite differences, only holding true for integral,ralues of the
variable (argument); secondly, and more essential, these authors obtained the
dietributirn of electrons and photons with respect to the energies in the shower,
thus solving only part of the problem.
In many works on the theory of showers, and in comparatively recent ones
01. Heisenberg, Koemische Strahlung, 1943), the authors employ an energy distribu-
tion s;;ecti-sm of electrons, calculated by Arley (Proc. Roy. Soo., 168, 519. 19 8,
and Arley, Mri2r3en, Danske Videnekabernes Selekab. 17, No 11, 1940) taking into
consideration tha ionization losses approximately. Meantime, Arley's spectrum
apyeare to be roughly inaccurate (as shown in section 14 of this book), and farther
study leads to a lower evaluation of the number of electrons with low ener;3ios.
In the works of Bhabha and Ohzk-abrxty (Proc. Roy. Soc. 181, 267, 1943, and
Proc. Ind. Aced. Sci. 15, 464, 1942) there is developtid their cascade theory whibh-
tek_~s into account ionization losses and arrives at an expression for the full
number of particles; this expression differs from the one obtained by Snyder and
Ferber, in seCLlon 14 ox TnLJ DCok .L in euuw. Uai .._. ., .z..
methoQ also leads to on underestimate of the number of low-energy particles,
althdh in lesser degree than Arley's calculations. The difference of B9babha's
and PhaLwabarty's resu).ts from those of Snyder's and Berber's is explained by
this underesttimate, :?. also the arrore of their conclusion pertaining to the
energy spectrum of the electrons.
In the work of Corben (l'h_ys. Rev. 60, 435, 1944), an attempt is made to con-
struct a cascade theory, with more accurate expressions for the cross aection of
pair-formats m in heavy particles taken into consideration. A criticism of this
xvrk is given in section 18 of Chapter V of this book.
Thus, in the cited works, the processes existing in the field of low energies
are considered either for serarate problems or roughly approximate.
Chapter 11 discusses the cascade theory for the field of high energies.
In Chapter III a problem is proposes to find the full solution of the basic
equations of the cascade theory, including radiational retardation, pair-formation
(whose cross-section is given in an asymptotic form, ho'd.ng true in the case of
complete screening), and ionization hares.
Rjr applying the Laplace-Mellin transformation with respect to the variable
E, the energy of a particle, and the Laplace transformation with respect to t, the
thickness of the layer penetrated, one can reduce these equations to an equation
in finite difference, which was successfully solved. The latter transformation
was first applied by the author in 1940. (DAR 33, 609, 19111). In addition, a
certain function entering this solution is replaced by an expression closely
approximating it in the variables' transformation region. In the cane where ioni-
tatitn losses are disregarded, the approximating function leads to a similar one
vhich is also accurate. By means of a transformation to the plans of complex
variables, the solution succeeds in presenting the form of a potential series in
terms of the small parameter ?DSO, where 6 is the so-called "Mitical" energy
(see Chapter II' and $o IN the energy of the original particle.
~p per,-, .bJ 7 7 ~~
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L
shall obtain a function which gives the dependenco of the full number of particles
upon the depth t of the layer, and also the energy distribution spectrum for any
depth t. If desired, it is possible to extract even succeeding terms of the
expansion; these give, however, for the majority of interesting cases, only
small corrections, that is, the first term is sufficient without the second, etc.
The full number of particles calculated in this manner are obtained in the
form of an integral in the complex-number plane. This integral is analogous to
the corresponding integral calt1,lated by Snyder and Berber. However, in place
of one of the integrands, which is determined by Snyder and Serber tniy for
integral values of the variable (argument), we have in our case established an
analytic fuA tion (complex) which assumes for integral values of the variable
(argument) the same values which are in Snyder's and Berber's function. Cbvioualy
this integral is calculated by Sommerfeld's method of "passing", given by the first
two nondisappearing terms in the expansion of the logarithm of the integrand function
in the form of Taylor's series, which corresponds to the first term of the expansion
of the solution in a potential series of the quantity (log to/,O )-1. Although,/Eo
is a small quantity, the logarithm of Eo/,I van not be very large. For showers rvrmed
'by the electrons of the atmosphere in lead, it is of the order of magnitude 5. There-
fore, during the calculation of the integral, we employed, in the expansion of the
logarithm of the integrand function, terms of higher order and the expression obbainsd
by us is thus accurate filly up to quantities of the order (log ;4 ' )-2.
In 19'39, I. To. Tatum and the author obtained an "equillbritm" electron-energy
distribution-spectrum which was 'neutralized' (averaged, etc.) with respect to the
total cascade curve. It was obtained as a result of solving the !xssic eOue.tions of
the thbory, taking into consideration ionization losses. (J. Phys, DSSR 1, 177, 1939).
In this research the emectram of delta-electrons and decay electrons was calculated.
m e oasis reeu.as o-- tune reaeercn are given in chapter lv, where the energy spectrum
of particles for various depths are also calculated. Furthermore, in this chapter
the average erargiee of the particles, as well as the logarithm of the average ener=
which is essential for a more accurate estimate of ionization looses 're calculated.
The obtained expressions are compared with the results of other authors, particularly
with Rossi's and E apman'e (Phys. Rev. 61, 414, 1942), which are calculated by means
of numerical integration. In this chspt,r, furthermore, the approximate value of
the influence of the Compton effect on the electrical distribution of electron and
photons are derived.
Chapter V is devoted to the theory of showers in heavy elements. In the theories
developed up to this time, the absorption coefficient for photons is assumed to be
equal to a constant, not depending upon the energy's magnitude. Moreover, for heavy
elements, particularly for lead, where the process of shower formation proceeds --1th
special . intensity, this assumption cannot be considered justified. Actually, the
absorption coefficient of photons in lead is not constant, but varies in the essential
region of energy variations three times.
It Chapter V the number c' particles at the maxinmxm of the cascade curve, and
the position of this maximum, taking into consideration the dependence of the absorp-
tion coefficient of photons upon energy are also calculated. The essentials of the
method are included, in the determination of the connection between the position of
the maximum and xumber of partiolea at the maximum, by the expression of the
following form:
I1 =
p r, :,Nd1
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e
d
work he also Lives a molhod of c4laulating the an egvare angle of e e o t?
the shower particles and Also the "width" of thr.lshvuer (mean square spatial
deflection).
xn... eor to the calculation of these magnitudes, the angle of deflection was
effect from lead to aluminium and iron is calculated; here it i? esbumed tha
layers of eluut