THE SOVIET JOURNAL OF ATOMIC ENERGY VOLUME 11, NO. 4

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP10-02196R000600070002-0
• Release Decision: RIPPUB
• Original Classification: K
• Document Page Count: 126
• Document Creation Date: January 4, 2017
• Document Release Date: September 25, 2013
• Sequence Number: 2
• Publication Date: April 1, 1962
• Content Type: REPORT

Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Volume 11, No, 4LEGIB
THE SOVIET JOURNAL OF
April, 1962
ILLEGIB
OMIC ENERGY
ATOMHasi
TRANSLATED FROM RUSSIAN
CONSULTANTS BUREAU
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
SOVIET
ANALYTICAL CHEMISTRY
^
ULM
A collection of ten papers from the Consultants Bureau
translations of the Soviet Journal of Analytical Chetvistry
and 'the famous "Dokladyn of the Academy of Sciences
(1949-58) ... This collection will acquaint the analytical
chemist working in this ?field With Soviet techniques for
the determination of uranium in solutions, in. ores ,and the
products of their treatments, and in accessory minerals,
plus methods for the determination-of-impurities in uranium.
heavy paper covers illustrated
$10.00
CONTENTS
? Extraction ofUranyl a7I\litroo- 8 -naphtholate and Sepa-
ration of Uranium from Vanadium and Iron.
? The Composition of Uranyl Selenite. A V,olumetric Method
of Determining Uranium.
? The Composition of the Luminescence Center of Sodium
Fluoride Beads Activated by Uranium.
? Rapid Luminescent Determination of.Uranium fill Solutions.
? Preparation of Slightly Soluble Compounds of Quadrivalent
Uranium Using Rongalite.
? Investigation of Complex Compounds of the Uranyl IonS
Which are Of 'Importance in Analytical Chemistry.
? Uranyl and Thorium Selenites.
V
? The Evaporation Method and Its Use forthe Determination
of Bpron and Other Impurities in Uranium.
? Spectrographic Determination of Uranium in Ores and the
Products Obtained by Treatment of These pres.
? Determination of Uranium in Accessory Minerals.
CONSULTANTS BUREAU
227 WEST 17TH STREET. NEW YORK 11. Y
[ Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
EDITORIAL BOARD OF
ATOMNAYA gNERGIYA
A. I. Alikhanov
A. A. Bochvar
N. A. Dollezhal
D. V. Efremov
V. S. Emel'yanov
V. S. Fursov
V. F. Kalinin
A. K. Krasin
A. V. Lebedin'skii
A. I. Leipunskii
I. I. Novikov
(Editor-in-Chief)
B. V. Semenov
V. I. Veksler
A. P. Vinogradov
N. A. Vlasov
(Assistant Editor)
A. P. Zefirov
THE SOVIET JOURNAL OF
ATOMIC ENERGY
A translation of ATOMNAY A ENERGIY A,
a publication of the Academy of Sciences of the USSR
(Russian original datedOctober, 1961)
Vol. 11, No. 4 April 1962
CONTENTS
PAGE
RUSS.
PAGE
Atomic Science and Technology and the Building of Communism. V. S. Em el 'y anov
947
301
Interaction of Charged-Particle Beams with Plasma. Ya. B. F a in ber g
958
313
Magnetic Traps with Opposing Fields. S. Yu. Luk 'yanov and I. M. Po dgornyi... ?
980
336
Physical Investigations in the Cyclotron Laboratory of the I. V. Kurchatov Institute of Atomic
Energy. N. A. Vlasov and S. P. Kalinin
989
345
A Survey of Nuclear-Reactor Design Methods. G. I. Mar chu k.
1000
356
The Future of Fast Reactors. A. I. Leipunskii, 0. D. Kazachkovskii, and
M. S. Pinkhasik.
1017
370
Some Results and Perspectives of Nuclear Radiation and Isotope Use in Russian Science and
Industry. P. L. Gru z in
1027
379
LETTERS TO THE EDITOR
t/T`he Elastic Scattering of Neutrons with an Energy of 15 Mev by Nuclei of Copper, Lead, and
U238. B. Ya. Guzhovskii.
1041
395
Measurement of the Cross Sections for Inelastic Interaction of Neutrons with an Energy of 13 to
20 Mev using Certain Isotopes. Yu. G. Degtyarev and V. G. Nadtochii
1043
397
The Inelastic Scattering of 14 Mev Neutrons by Sodium, Iron, Nickel, and Lead Nuclei.
V. I. Sukhanov and V. G. Rukavishnikov
1044
398
The Attenuation of Neutron Flux in the Reinforced-Concrete Shielding of a Synchrocyclotron.
M. M. Komochkov.
1046
399
The Long-Lived Isotope A126 in the Aluminum used in the Construction of a Nuclear Reactor.
S. S. Vasil'ev, T. N. Mikhaleva, N. P. Rudenko,
A. I. Sevast'yanov, and V. S. Zazulin
1048
401
BIBLIOGRAPHY
Review of Gosatomizdat (State Atomic Press) Publications for 1960 and 1961
1050
404
Note to subscribers? The author index for volumes 10 and 11, 1961 will be published in volume 11, no. 6.
Annual subscription $ 75.00
Single issue 20.00
Single article 12.50
@ 1962 Consultants Bureau Enterprises, Inc., 227 West 17th;St., New York 11, N.Y.
Note: The sale of photostatic copies of any portion of this copyright translation is expressly
prohibited by the copyright owners. ,
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
ATOMIC SCIENCE AND TECHNOLOGY AND THE BUILDING
OF COMMUNISM
V. S. Emel'yanov
Chairman of the Government Committee of the Council of Ministers, USSR,
on the Use of Atomic Energy,
Translated from Atomnaya Energiya, Vbl. 11, No. 4, pp. 301-312,
October, 1961
Original article submitted August 31, 1961
In the project for the new Program of the Communist Party of the Soviet Union it is stated that" . . . the
development of new technology will be used for the radical improvement and alleviation of the conditions of work
of the Soviet man, for shortening the working day and providing a more convenient existence, for putting an end
to heavy physical labor, and then to every kind of unskilled labor."
The progress of physical science, and, in particular, of nuclear physics opens up great new possibilities for
achieving the goal set? building Communist society.
This new powerful source of highly concentrated energy? atomic energy ? can considerably increase the
amount of energy which the Soviet man has at his disposal, and can enable us to solve problems which it is practi-
cally impossible to solve with the usual sources of energy. Radioactive isotopes as a source of radiation offer note-
worthy possibilities for the automation of industrial processes, then for replacing manual labor by the work of
machines. Isotopes are being used more and more widely in medicine, scientific researches, and the most varied
branches of the national economy.
Scientific research and investigation in the field of nuclear physics stands at various stages, and the scale of
work on using the results is also different. Up to the present time the work that has been advanced further than any
other is that in the field of using the fission energy of heavy nuclei ? uranium and plutonium. These processes
have found practical application ? they are the basis for the design, operation, and building of atomic electric
stations and power units.
The first atomic electric station of industrial type in the world,with an output of 5000 kw, was put in opera-
tion in the Soviet Union more than seven years ago, and from that time has operated successfully and without in-
cident.
Instructions being completed on the first series of large atomic electric stations at Voronezh and in Beloyarsk.
The uranium-graphite reactor of the I. V. Kurchatov Beloyarsk atomic electric station, with an electrical
output of 100,000 kw, is an original type of construction, developed by Soviet specialists. This reactor, using
slightly enriched uranium,produces superheated steam at a pressure of 100 atm. Thus, in its operating character-
istic the Beloyarsk reactor will be the best in the world from the point of view of present-day energy production.
The Novo-Voronezh atomic electric station uses water-water-reactors, each of which is designed for an
electrical output of 210,000 kw.
Experimental atomic electric stations are being constructed in Czechoslovakia and in the German Demo-
cratic Republic with the aid of the Soviet Union. At the atomic electric station in Czechoslovakia, with a power
of 150,000 kw, a reactor is being constructed in which the moderator is heavy water and the coolant is carbon
dioxide gas. In the German Democratic Republic the output of the first unit of the atomic electric station, using
a water-water reactor, is 70,000 kw.
The work of Soviet scientists and engineers in the field of atomic energy is accompanied by the construction
of many research and experimental nuclear reactors and assemblies of different types and power levels. These in-
clude reactors using as moderator: graphite, heavy and ordinary water, organic liquids, and as coolant: ordinary
947
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
and heavy water, organic liquids, fused metals, and other coolants. Reactors have been constructed working at
various neutron energies and with various neutron flux densities in the steady and pulsed state.
As we know, the Soviet Union has enormous resources of organic fuel and water power, which are able to
supply the country for quite a long period, although there are isolated regions in the Soviet Union which are con-
siderably removed from the usual sources of energy. In the project of the new program of the Communist Party of
the Soviet Union, which will be considered at the XXII Congress, it is stated: As the production of atomic energy
becomes cheaper, the construction of atomic electric stations will be expanded, especially in regions lacking in
other energy sources . . . ".
From the design, construction, and operation of all these and other atomic electric stations with reactors of
various types and powers, a large amount of experience will be collected, which will make possible analysis and
engineering and economic evaluations leading to new paths of action and the building of even more perfected in-
stallations.
In particular, our scientists have constructed one of the most promising reactors at the present time, the ex-
perimental fast-neutron reactor, which reached criticality in June, 1958 and has already been operating successfully
for more than three years. Reactors of this type make it possible to get an exchange for each kilogram of U 2* or
plutonium "burned up up to 1.5 kg of plutonium or U 233 from the improved neutron balance. This opens up the
possibility of considerably more complete use of nuclear fuel resources.
While at the present time the usual thermal-neutron reactors use up only 0.4-0.5% of the uranium supplied,
?
i.e., up to 5 kg of each 1000 kg, and thorium can be used as a nuclear fuel only at the cost of consuming a con-
siderable quantity of U235, in fast-neutron reactors it is possible to use up both the U 2* and the thorium com-
pletely. With nuclear reactors of this sort it is possible to build atomic electric stations with large output and a
comparatively small yearly uranium consumption.
Calculations show that it is possible to build atomic electric stations with a total electrical output, for
example, of 100 million kw, which, using fast-neutron reactors, will consume less than 1000 tons of natural uranium
per year. It is well known, that operating an ordinary coal electric station at a total output of 100 million kw
requires a yearly consumption of 200-300 million tons of coal.
Up to the present time, considering reactors giving extensive breeding of the fissionable material and com-
plete utilization of uranium and thorium, it may be concluded that from the scientific standpoint we are dealing
with the problem which has been fundamentally golved and we are talking about using the fission chain reactions
of heavy elements ? uranium and plutonium ? to produce large quantities of electrical energy. Atomic electric
stations using reactors providing extensive breeding of nuclear fuel have not yet been tried out in fact, and the
problem of producing large quantities of electrical energy from the nuclear fission reactions of uranium and plu-
tonium has not yet been worked out in many of its engineering and economic features. Work in this direction is
being carried out here, as well as in the USA, England, and France.
The initial expenditures of fissionable material for loading fast neutron reactors are still large. It is necessary
to accelerate the conversion of nuclear fuel, as well as solve other technological problems.
Further automation and refinement of the processes for extracting and purifying plutonium or U233 from fission
fragments, and automation of the production of fuel elements along with the development of economically feasible
reactor designs, are some of the more important problems standing in the way of wide use of nuclear fuel for the
development of a large atomic energy industry.
Reactors giving extensive breeding from nuclear fuel combined with other types of reactors should be care-
fully studied in relation to their application to the problems of electrification of the whole country and the wide
industrial use of atomic energy.
Along with the developments in the construction of nuclear reactors for electric stations, as well as for re-
search purposes, Soviet specialists have designed reactors for use in ship installations. The Soviet Union, which
possesses first class atomic submarines guarding the water approaches to our country, has devoted considerable
attention to the peaceful uses of atomic energy in the naval fleet. In 1960, the flagship of the Soviet icebreaking
fleet, the atomic icebreaker "Lenin," made its first voyage on the northern sea route.
948
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Fig. 1. Mounting of headers for the fuel channels on the top plate of the reactor of the I. V. Kurchatov
Beloyarsk atomic electric station.
Some data is given below on the Soviet atomic ship as compared with the first American atomic commercial
ship 'Savannah."
Icebreaker
"Lenin'
"Savannah'
Date put into service
Displacement,thousands of tons
Propulsion power, thousands of hp
1959
16
44
1961 (Planned)
21.8
20
949
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Fig. 2. Experimental fast-neutron reactor building, constructed in 1958.
Fig. 3. The atomic icebreaker 'Lenin" with a convoy of ships in the ice floes of the Arctic Ocean.
For the first time in the history of the Arctic a convoy of Soviet ships led by the atomic icebreaker *Lenin?
moved out of the Kara Sea east to the Laptevs Sea. The icebreaker ?Lenin ? led 92 ships over the northern sea
route through the Arctic ice floes, when the ice had reached a thickness of 2.5 meters.
950
[Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
?.;
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Thus, we have had our first experience in using atomic energy in an icebreaker, which confirms the results
of the diverse work done by Soviet specialists in nuclear reactors for ship installations.
One of the important problems in the wide development and use of atomic energy is the problem of radio-
active wastes. In the production of plutonium and the reworking of the fuel elements from atomic electric stations
and power installations a large quantity of radioactive wastes is formed, which decompose only partially in a short
period of time, while a considerable fraction remains dangerous to man for many decades. Some of the radioisotope
fragments are beginning to find practical application in industry, medicine and other fields. However, such use has
so far has not been on a considerable scale.
In our country, the radioactive wastes from the purification of fuel elements spent in the reactor are kept in
special containers. In the USA and England, containers of radioactive wastes from plutonium production are dis-
carded in seas and oceans, creating a potential danger from radioactive contamination of the plant and animal world
of the seas and oceans, as well as a potential danger for human beings. Radioactive substances accumulate in plant
and animal organisms, part of which is used for human consumption.
The studies in the field of biology and medicine and,in particular,the genetic side of this question indicate a
detrimental effect of radioactive contaminants on human beings and require the development of special means to
prevent the penetration of radioactivity into the human organism.
Fig. 4. Pulsed fast-neutron reactor of the United Institute of Nuclear Studies in Dubna.
951
I Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Soviet scientists devote a large amount of attention to the study of radiological problems connected with all
possible radioactive contaminations. Searches are being made for ways of burying radioactive wastes which will
give a reliable guarantee that never, neither at the present time nor in the future, will this radioactivity get out of
control, nor lead to infection of the surroundings, nor bring harm to man.
In order for the atomic energy industry to play a
material role in the over-all energy industry of the
country, in addition to the solutions of the problems
indicated, there must be a considerable reduction in
the inherent cost of the electrical energy developed in
atomic electric stations.
One of the greatest scientific problems in the
field of atomic energy is the direct transformation of
the energy of nuclear processes into electricity.
In the Soviet Union,work on the direct transforma-
tion of heat into electricity has been going on for a
long time. Academician A. F. Ioffe has not only worked
out the theory of thermoelectric converters, but under
his direction the first thermal batteries were constructed.
New semiconductors, developed by Soviet scientists,
have even higher working coefficients, and retain their
properties for a long time.
In a nuclear reactor the energy may be produced
at a very high temperature, which gives the promise of
high efficiencies in the transformation of fission energy
into electricity. Direct transformation of nuclear energy
into electricity considerably simplifies the engineering
scheme of producing electrical energy and will have
tremendous importance in many branches of technology.
Therefore, in the project of the new program of the
party, there is a direct statement of the need to extend
the work on methods " . . . of direct transformation of
heat, nuclear, solar, and chemical energy into elec-
tricity . ?" It is obvious, that with the great promise
awaiting the atomic energy industry, special attention
Fig. 5. Exhibit of protective clothing used in monitor- should be given to the physics of nuclear reactors.
ing radioactive contamination (Pavillion "Atomic Soviet scientists are carrying on various types of
energy for peaceful purposes" VDNICh, 1961). work in this field, and will extend them systematically
to provide a basis for industrial development. As an
example, we can indicate one of the directions of this
work.
In 1960, the United Institute of Nuclear Studies in Dubna began to operate a new Soviet nuclear reactor in-
tended for studies in the field of neutron physics, which is distinguished by a good deal of originality. This is the
only reactor in the world using plutonium rods and a disc of U235 rotating at 5000 rpm. At the pulse maximum the
reactor power reaches 3000 kw. The reactor makes it possible to obtain periodically, 8.3 times per second, an
over-all neutron flux equal to 1017 neutrons/sec at the pulse maximum.
In the past few years, in addition to the use of atomic energy for power installations, the use of radioactive
isotopes in nuclear radiations has been greatly developed.
952
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
. At the present time, radioactive and stable isotOpes and nuclear radiations are being used in our country by
more than 2500 research, medical, and industrial organizations. , Now, radioactive isotiopes are being used in practi-
cally all branches of the national economy, although to different degrees. The industries of the Soviet Union now
produce more than 300 radioactive and stable isotopes, and from them prepare sources of radiation and labelled
chemical compounds. 1
1
Isotope production is widely distribuied inside the country and exports are made to the People's Democratic
Countries, as well as to Japan, the United Arab Republic, Iraq, Mexico, and other counitries.
? Fig. 6. Demonstration Hall of the Moscow "Isotopes" Store.
However, the practical use of isotope Methods is really only beginning. It will be continuously enlarged:
The new radical force of nature must be employed to.full. It should be noted in this connection that at the present
time only one-tenth of the, radioactive and stable isotopes known to science are being used for practical purposes.
The use of radioactive isotopes as tracer atoms for scientific investigations and many other purposes is firmly
entienched ikpractice. Methods have also been widely developed based on the penetrating power of ionizing radia-
tions and their destructive action, and on energy release on the activation of other materials, etc.
The various devices based on the use of radioactive isotopes to a high degree meet present-day requirements ,
of industry, which is characterized by a-rapid increase in the rate of flow of various processes, a transition to high
temperatures and pressures, and by the use of continuous automatically-regulated processes. Thus, for example, in
heavy, and light metallurgy, radioactive isotopes and ionizing radiations make it possible to have continuous monitor-
ing of the mixing of metal during crystallization, and to create a technology of continuous monitoring of the thick-
ness of rolled sheets and plates, and the loading of layers of materials in blast furnaces. In the construction of.
machinery, isotope methods are used in examining the product for defects, and observing the wear of machine and
instrument parts. In construction work,isotope methods are used to monitor the quality of concrete structures while
the concrete is being laid and checks can be made on reinforced concrete structures.
Radioactive methods are firmly entrenched in geologic prospecting practice as a means of searching for and
prospecting useful ores. From data of the Institute of Economics of the Academy of Sciences, USSR, in the oil fields
of Azerbaijan and western Ukrainia alone, more than 2.2 million tons of high quality petroleum have been found
953
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
by tracing out neglected deposits with these methods. From the same data, the economic effect of the use of radio-
active methods in prospecting and exploiting useful ores amounted to more than 45 million rubles in 1960.
Great possibilities are opened up by the use of ionizing radiations to produce various radiation chemical
reactions.
As an example of reactions which are economically important, we can already mention the radiation oxida-
tion of organic compounds, benzene in particular, and the radiation polymerization of a number of organic com-
pounds, which makes it possible to prepare materials with substantially improved and new properties. The radiation
synthesis of polyethylene makes it possible to carry on the reaction at a pressure of 250-300 atm and a temperature
of 30-80? C without the use of catalysts. The use of insulation made out of thermally stable polyethylene in
electrical engineering makes it possible to reduce the copper required two or three times because of the larger
current densities that can be used without shortening the life of the conductor.
These and many other fields where radioactive isotopes are used show that radioisotope technology is an im-
portant method in the development of technology in the national economy.
At the present time in the Soviet Union radioactive and stable isotopes are being used to prepare more than
700 different chemical compounds.
Great possibilities for reelective isotopes in nuclear radiations are opened up in medicine, and they find wide
application in the diagnosis and treatment of a number of diseases. For example, radioactive isotopes may be used
to study the functions of various organs and systems of the body without disturbing their integrity. Thus, the isotope
I131 is used for a diagnosis of diseases of the thyroid gland. The isotope P32 makes it possible to determine the
quantity of blood circulating in the organism. Then in the diagnosis of tumors of the central nervous system in the
brain, use is made of the radioactive isotopes of radon, xenon, and iodine. A technique has been worked out for
external irradiation in the tele-gamma-apparatus using Co6? and CO" for the treatment of cancerous diseases of
the skin, the esophagus, and the lungs, as well as other diseases.
Radio surgical methods have been worked out for radiation therapy inside the cavities and tissues of the body,
which are used in combination with external irradiation. For this purpose, use is made of the isotopes Co60, Cs137,
and Au l" in the form of a colloidal solution, granules of Y",etc.
The examples given of the use of radioactive isotopes and radiations in medicine only partially illustrate
the enormous possibilities of the peaceful uses of atomic energy for the welfare of man.
A great contribution can be made by the use of isotopes in agriculture. New possibilities and prospects are
to be expected in the future, in particular, the use of radioactive radiations for preserving agricultural products.
Present-day technology has a large range of radioactive isotopes at its disposal, with different energies,
different half-lives, and other special properties. This makes it possible for the technologist to solve problems
which could not even have been dreamed about previously. For example, the use of radioactive isotopes in auto-
mation of production may become a powerful means of solving one of the problems presented in the plan of
the new program of the Communist Party of the Soviet Union: " . . . putting an end to heavy physical labor, and
then to every kind of unskilled labor." Thus, for example, in the mining and smelting branches of industry in the
grinding, transportation, cleaning and smelting of ores and coal, where simple monotonous operations require a
man to be in ore and coal dust, in the midst of the clamor of operating machinery, the use of equipment with radio-
active isotopes makes it possible to automate the production processes completely and replace the labor of human
beings with the work of machines. The first production experiments on the use of radioactive apparatus in the
Krivorozh'e South Mining and Smelting Combine and at the Slantsy Mine in the Estonian SSR show that the method
is completely possible, and open up one more method of using radioisotopes.
The use of isotopes in the coal industry for automatic stabilization of the motion of the coal combine at
the "coal-sorting" junction makes possible constant automatic control of the position of the cutting instrument
and thus makes complete use of the speed capabilities of the combine. This is a trend which may lead subsequently
to the complete automation of subterranean work with the control of the machinery transferred to the surface.
It is clear from what has been said that atomic energy is now finding practical application not only for war-
like purposes but in various branches of the national economy as well. It has come out from behind the walls of
the laboratories into wide ranges of industrial application. After the atomic electric stations and reactors which
954
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Fig. 7. General view of the cyclotron for accelerating multiply-charged ions.
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
have been built and are being built have been tested in operation, after the work of construction and evaluation of
various types has been completed, and it has been found out which are the best, the most economical and the most
reliable types of equipment, wide prospects will be opened up for the construction of electric stations and power
installations, and radioactive isotopes and radiations will find even broader application in the national economy.
The foundation of our work on the use of atomic energy is its powerful scientific basis, the development
and enlargement of which has received a large amount of attention from our Communist Party. This basic scientific
structure ensures, in particular, that new scientific and practical prospects will be opened up.
The scientific treatment of the various problems of nuclear physics is already at the present time showing us
now even more attractive possibilities with use of the energy from nuclear reactions.
The use of atomic energy from the fission of heavy nuclei is not the only possible way of using the energy
hidden in the atomic nucleus. Here, above all, attention should be called to the problem of controlled thermo-
nuclear reactions, the solution of which is one of the more important tasks assigned by the Communist Party of the
Soviet Union.
Academician I. V. Kurchatov, as early as 1956, appearing at the Twentieth Congress of the Communist Party
of the Soviet Union, pointed out how large a role would be played in science and economics by the solution of the
problem of the controlled thermonuclear reaction. He spoke of the fact that in the hydrogen bomb we already know
how to create the conditions necessary for the fusion of hydrogen nuclei, i.e., effect thermonuclear synthesis, but
we must now control this reaction in such a way as to avoid an explosion.
In the USSR work on controlled thermonuclear synthesis is being carried out on a wide front. To carry out
the research in physics a number of large installations of various types have been created, including the very large
'Ogre experimental installation with a vacuum chamber 1.4 meters in diameter and 12 meters long. Several in-
stallations are in the process of construction. We are convinced that the problem of practical utilization of controlled
thermonuclear reactions will be solved.
In the work on thermonuclear synthesispreactions are being studied which occur at temperatures of millions
of degrees, where matter becomes plasma, a new, little-studied state. A new field has been added to physics ?
plasma physics. The development of this field is of fundamental importance, since, if the work on thermonuclear
synthesis is successfully completed, it will completely cover every energy requirement of the whole population of
the terrestial globe for an unforeseeable length of time. This work is also important because of the fact that the
developments are sure to uncover a number of new scientific results and data important in practice. Thus, in plasma
work we must have a technique of producing very high vacua (1040 - 10 -11 mm Hg) in large volume. Experience
in this field is of value to a number of branches of the national economy. Plasma work entails the production of
powerful magnetic fields (50,000 - 200,000 oersteds).
Recently, as a result of the researches of Soviet scientists, intermetallides have been discovered which show
superconductivity and make it possible to produce powerful magnetic fields at practically realizable helium tem-
peratures, which, naturally, is of considerable importance for other branches of science and technology.
But nuclear physics is still far from exhausting all its possibilities. Thus, studies on the atomic nucleus have
led scientists to discover new atomic particles, the antiparticles. In recent years, for example, more than ten
antiparticles have been discovered. The majority of them have a lifetime less than a hundred thousandth of a
second, while some of them are stable or have a long lifetime.
Data on some of the antiparticles is given below:
Mass
Mean Lifetime
Year of
discovery
Antiproton
Antineutron
Positron
938
940
0.511
Stable
Stable
1.013- 10 3 sec
1955
1956
1932
Union of the proton with its antiparticle, the antiproton, produces annihilation, accompanied by an ex-
tremely high yield of energy, approximately 1000 times greater than in fission of nuclei or in thermonuclear syn-
thesis. The problem of particle annihilation has serious scientific importance, and treatment of the problem will
give a more profound understanding of the structure of matter.
956
[ Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
A powerful means of studying the structure of the nucleus of the atoms of matter is to study the effect on
nucleus of other nuclei and particles raised to high energies by means of powerful accelerators. This is the technique
in most universal use at the present time, and in the coming years it will continue to develop. Thus, in the near
future new powerful accelerators will be put into service. Recently, construction was started on the most powerful
accelerator in the world with sharp focusing at a nominal proton energy of 50-70 Bev. The more important param-
eters of this accelerator are as follows:
Nominal proton energy, Bev
50-70
Mean radius of orbit, meters
236
Maximum field, oersteds
10,000-12,000
Injection energy, Mev
100
Number of magnets
120
Total magnet weight, tons
> 20,000
The new accelerator installations still further strengthened the material basis of Soviet research on the atomic
nucleus.
Considerable scientific interest is presented by the cyclotron for accelerating multiply-charged ions, recently
constructed at the United Institute for Nuclear Studies in Dubna. At the beginning of September, 1960, the cyclotron
produced the first beam of accelerated particles, and at the present time the accelerated ions have reached an energy
and intensity sufficient to begin experimental work on the study of reactions in complex and heavy nuclei. However,
I
even wider prospects are opened up by new methods of accelerating particles, which provide practically unlimited
possibilities for interaction with matter, thus broadening our understanding of the struCture of matter.
Our whole country comes to the Twenty-Second Congress of the Communist Ply of the Soviet Union with
great achievements. There are substantial advances in the field of utilization of atotiric energy as well. Our
achievements form the basis for multiplying our efforts in the progressive movement ahead.
Having methods of utilizing atomic energy on an economical basis means creating a new
which is the economic basis of a new society which will realize the great principle 'from each
abilities, to each according to his needs."
A greater supply of energy means higher productivity of labor which is the decisive factor
the new social structure ? Communism. The grandiose scale of this task inspires all workers in
utilization of atomic energy to unrelenting labor in the name of a glorious future.
We recognize that we are only at the threshold of a new era, we recognize the whole importance of our
efforts for the movement of our country ahead toward communism, and we shall apply all our energies ". . . to
fortify the leading positions won by Soviet science in the more important branches of knowledge, and to occupy a
leading position in world science in all fundamental directions ."
energy industry,
according to his
in the victory of
the field of the
957
I Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
INTERACTION OF CHARGED-PARTICLE BEAMS WITH PLASMA
Ya. B. Fainberg
Translated from Atomnaya fnergiya, Vol. 11, No. 4, pp. 313-335,
October, 1961.
Original article submitted July 28, 1961
The interaction of charged-particle beams with plasma is of great importance in various kinds of gas-discharge
devices used in research intended to achieve CTR. ? Beam-plasma interactions are also of great importance in new
methods of accelerating charged particles and in plasma devices for amplification and generation of microwaves.
In spite of apparent differences and the rich variety of interaction mechanisms between beams and plasmas,
there are only three basic processes involved in these interactions: the Cerenkov effect, the Doppler effect
(anomalous and normal), and plasma polarization produced by the passage of a charged particle through the plasma.
There is another mechanism, which operates when charged particles or oscillators move through a bounded or
spatially-periodic plasma This is the so-called parametric Cerenkov effect.
In most cases the plasma density no is relatively small (n0 1012 - 10") so that the energy lost via these
mechanisms for an individual charged particle traveling a unit distance d6/dx is insignificant, being of the order
of 10-8 - 10-5 ev/cm. In most cases, however, a beam of charged particles interacts with the plasma. In this case
the effective strength of the interaction increases markedly [1] because the self-modulation of the beam produced
by the interaction means that in the final analysis we are dealing with a coherent interaction between a beam of
charged particles and a plasma [2].
The energy loss of the beam particles due to oscillations is quite significant and can be as high as 108-104
ev/cm per particle when the number of particles in a bunch is N 107 - 108. Because of the appreciable intensity
of the interaction between beams or charged-particle bunches and a plasma it is reasonable to assume that these
interactions are responsible for a number of the effects observed in gas discharges. It would appear, for instance,
that these interactions lead to many plasma instabilities, produce a Maxwellian distribution in the absence of
collisions, and affect various transport phenomena (conductivity, diffusion) in a plasma.
The high interaction energy of charged-particle beams or bunches in a plasma (compared with the interaction
energy for an individual particle) can be exploited in plasma injection in magnetic-mirror systems, plasma heating,
in measurement of plasma parameters, in the determination of distribution functions, and for various other pur-
poses.
As far as CTR systems are concerned, the most important aspect of this problem is the investigation of con-
ditions associated with the production and suppression of instabilities due to the interaction of charged-particle
beams with a plasma.
The motion of a beam in a plasma is said to be unstable if any initial perturbation (fluctuation) of the beam
or plasma tends to grow. A necessary condition for the appearance of an instability is that the conditions corre-
sponding to at least one of the basic mechanisms considered above (Cerenkov effect, anomalous or normal Doppler
effect) be satisfied. Inasmuch as radiation processes can be accompanied by absorption processes, the number of
particles in the beam that give energy to the electromagnetic field must be greater than the number of particles
that absorb energy from the field." There is one other condition that must be satisfied before an instability can
arise: Particle bunching must occur in a phase region in which particles lose energy to the electromagnetic field.
In a number of cases this requirement on particle bunching is satisfied automatically if the radiation condition is
satisfied for an individual particle. The electromagnetic fields due to radiation by particles in the beam cause
bunching, i.e.. "self-modulation" of the beam; in turn the increasing modulation of the beam causes an increased
? CTR ? controlled thermonuclear reactions.
? * This requirement imposes certain conditions on the unperturbed velocity distribution functions for the particles in
the beam and in the plasma (see below).
958
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
radiation intensity, because particle bunching enhances coherent radiation. Hence, in its initial stages the growth
of an instability is exponential.
From the point of view of quantum theory [3], an instability implies that because of greater populations in
"upper" energy levels (beam particles) there are more transitions in which induced emission occurs than in which
absorption occurs. For example, in a beam of free particles the 'upper' levels are associated with particles of
higher velocities. Hence, an instability arises if the derivative of the distribution function for particles in the beam
is positive at velocities for which the particles can interact strongly with the plasma. This result also follows from
classical considerations.
1. Theoretical investigations of the excitation of longitudinal oscillations by a beam of charged particles have
been carried out by A. I Akhiezer and the author, and by Bohm and Gross [1]; thesei investigations pointed up the
importance of beam-plasma instabilities and showed that the instability becomes particularly strong if the ordered
beam velocity Vo becomes greater than the thermal velocity of the plasma electrons VTe, i.e., Vo > VTe. The
elementary mechanism in this interaction is the Cerenkov effect [4] for the longitudinal plasma waves. When
ni. ? no the spectrum of excited high-frequency oscillations is determined by the Cerenkov condition and lies in
Th (no, )
co Vs
the frequency region ,-, coo! e growth rate of the instability 6 = Inico is rather large, 6 ,.,_ coo --?-
in this case. I no
Similar effects are found in the interaction of two or more beams [5]. For example, when two electron beams
(Maxwellian distribution of ordered velocities) interact [6], instabilities can arise if the Condition I Vol - V02 I > VT ei +
+ VTe2 is satisfied, where Vo and VTe are respectively the ordered velocity of a beam and the thermal velocity of
the electrons. The growth rate is also quite high for these instabilities. For certain beam parameters the conditions
for the longitudinal Doppler effect can be satisfied and the excited frequencies are given by (I)
VO2
I
V? ?175 yr,
Further investigations of excitation processes in plasma, carried out by G. V. Gordeev [7] have shown low-
frequency oscillations and ion waves (in the absence of a magnetic field) can be excited if VTe P 1/2 < Vo < VTe,
that is to say, these modes can be excited at low ordered velocities and consequently, low currents.
A large number of papers concerned with the interaction of charged-particle beams and "oscillator' beams
with plasma have been published in the last several years. This work has revealed the existence of a large number
(about 20) of different kinds of instabilities.
Most of these instabilities can be classified into three main groups corresponding to the three basic excitation
mechanisms:
1. instabilities due to the Cerenkov interaction of electron and ion beams with a plasma in a magnetic field.
These instabilities arise when the ordered beam velocity Vo is equal to the phase velocity of the wave Vph
and lead to the excitation of low-frequency and high-frequency oscillations. The low-frequency oscillations in-
clude ion-acoustic waves, Arfven waves, and magnetoacoustic waves; the high-frequency oscillations are the longi-
tudinal electron oscillations in the magnetic ?field.
Low-frequency oscillations can also be excited by drift currents in an inhomogeneous plasma [8, 9]. In this
case the drift motion plays the role of the ordered motion and the instability condition is given by Vdr = Vph (here,
Vdr is the drift velocity).
2. Instabilities due to the anomalous Doppler effect.
This effect requires that the velocity of the radiating particle be greater than the phase velocity of the plasma
wave (V > V h). In this case the radiated frequencies are determined by the relations co - k3V0 = -or
wres
cores
(here, co
- res is the natural frequency of the radiating particle in its own rest system and k3 = --
Vo Vph
1
Vph ,
" A table of the notation used in the present work is given at the end of the paper.
959
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
is the projection of the wave vector in the direction of motion of the beam). The excited frequency computed in
the reference system in which the beam is at rest coincides with the resonant frequencies of the beam, in particular,
the Langmuir frequency wo (in which case we have a longitudinal Doppler effect), or the Larmor frequencies WJj
(w = wo; = num; n = 1,2 . . .
An important feature of the anomalous Doppler effect, first pointed out by V. L. Ginzburg and I. M. Frank [3]
is the fact that the radiation process is accompanied by transitions to higher energy levels. Hence, it can occur
only in unperturbed oscillators, in particular a beam of charged particles with no initial transverse energy in a plasma
in a magnetic field. The conversion of freely moving particles into radiation oscillators takes place at the expense
of the energy associated with the longitudinal motion.
Instabilities due to the anomalous Doppler effect include those associated with the excitation of ion-cyclotron
waves in a plasma by electron or ion beams. The anomalous Doppler effect also causes high-frequency electron
oscillations (frequencies ? um; I 4 + (41 ); however the growth rate is smaller than for oscillations caused by the
Cerenkov effect. In all the cases cited above the oscillation frequency in the beam reference system is close to the
beam Larmor frequency. When plasma oscillations are excited by a high-density beam it is also possible for the
longitudinal Doppler effect to operate, in which case the oscillation frequency in the beam system is close to the
Langmuir frequency of the beam coo. In this case the oscillation frequency is (Jo wo ? 14 .
3. Instabilities due to the normal Doppler effect.
If electrons in a beam in a magnetic field have initial transverse energies, in moving through a 'plasma such
a beam can cause an instability due to the normal Doppler effect. An important feature of this instability is the
fact that it can arise when the beam velocity Vo < V h*. in particular, when Vo = 0. Thus, radiation due to the
normal Doppler effect is always possible. The only condition necessary for the appearance of this instability is that
the particle bunching in the phase region in which the particles lose energy to the electromagnetic field be stronger
than in the phase region in which the particles absorb energy from the field.** A mechanism for this bunching '
effect has been suggested by A. V. Gaponov [10]. This mechanism is based on the energy dependence of the os-
cillation frequency of a particle.
When an oscillator moves in a spatially inhomogeneous field and the oscillator frequency w wres ? the mean
displacement of the oscillator in a time large compared with the period of the oscillations is different from zero
and depends on the initial phase of the oscillator. This effect leads to spatial bunching of oscillators in a wave
field.
In Tables 1-3 we show the basic features of instabilities due to the Cerenkov effect, the anomalous Doppler
effect, and the normal Doppler effect. ???
2. These tables show that instabilities can arise at relatively low electron velocities (V0> VTe 11 1/2 and
Vo a- VA).
A reduction of the electron velocity for which an instability arises means essentially a reduction in the mini-
mum current necessary for exciting an instability; this current value can be smaller than the minimum current for
hydrodynamic instabilities. We may note, however, that the minimum current is not only determined by the critical
velocity of the ordered motion, but also by the electron density that is involved. For this reason, a number of in-
stabilities associated with high values of the ordered velocity can actually occur at low current values because the
instabilities are excited at relatively low-ordered electron densities.
? The frequencies radiated in the normal Doppler effect are given by co - k3 Vo = wres.
** Resonance ("currentless" or "beamless") methods of heating based on the ordered motion of ions across a mag-
netic field also lead to the development of low-frequency (co = nil) and high-frequency electrostatic instabilities.
*** Tables 1-5 are for purposes of illustration only. They contain the simplest and most striking cases. The
appropriate relations for instabilities in more general cases are given in the papers to which reference is made
in the tables. These tables are based on all pertinent work on this problem known to the authors. Both the
general cases and the cases given in the tables are considered in work to which references are made.
960
- Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
TABLE 1. Instability Due to the Cerenkov Effect.
c
Exited
oscillation
Excitation condition
Frequency .
spectrum, - Growth rate
hnudReco
Example
Imw)no=10
ni
vo= 3 '109
g,--T
2.
Longitudinal
Waves (low
V T
170>ITTe. --2- < (12-1-)1/ a '
100 1/-:--3 ( ni-V2 .
4.103 sec-i;
density
electron
beam)
' Vo 120 '
Vn. ( ni V/2
Vo t70 )ni
24/3 .no ) '
V4
ITC
' I
5.108 sec-i
[1] .
Ion acoustic
waves
1/2
1/2 < (
< Iro
m
k (T. , "112 .
M )
?
Imco=cooi . '
( 2Te
vo \112
2.108 sec-4;
[7]
x,(1-1-k2)2D)-1/2
.
x ? \. 3M fi
V'o
2Vre ?
!
lie
[1.11
Ion acoustic
waves in a
magnetic r
field
>>
.1, ( TMe Nt/2
)
x [1+ k 121X2D
-I- k? Te ' -112
j_ --7-d
>> 1
>>
[14
Arfven
waves
a) ordinary wave :
vg> v2A (1+ '?Lni ) ;
b) extraordinary wave.
Va cos20> VA (1+1-v: )
? kVA cos 0;
? [ v o2 ?1 ____ no i 2
v2
A nt .
1174- cos2 0-1 ?r-1211/2
t? A ? nt
113]
(14]
[151
[161
Longitudinal
waves in
? magnetic
e c _
le
tron beam
-1/T3
0
kl H .0
kV o ,coo I cos 01;
Vol/ -4-4 shot-
(44 n, cot2
4
2/a o ,
[15]
'9? 7112)
H < k (p 3. V 0 < 1/ (Oh 4. al cos2 0
(no 1 cos 0 I
175 ni s,
[16]
.144;
)1
24/3 no
(
,
?
? T is the temperature in degrees.
961
I Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
TABLE 1 (Continued)
Excited
oscillation
Excitation condition
Frequency
spectrum
Growth rate
ItncaRew
Example
=1012;n=1081
ve 3 ? 108
Literatu
reference
Longitudinal
waves across
magnetic
field (ion
beam
(30, tr/2)
'
koVo =(to}/QH)1/2
(041QH)1/2
V5 ( rt1 "8
3.107sec -1
(17]
24/8 no
Magneto-
acoustic
waves
(ni VT
P -
co < 1c3V0
Imo) --= ti1/2(00
1.5.109 sec '1
[24]
Ion-cyclotron waves
produced by an electron
beam
.0)ff ,----- koVo
?QH
0, vg , A
0)
3.104,sec-1
[17, 15]
? --, ll k 1 4- cos2
20)0 Vit
Ion-cyclotron waves
produced by an ion beam
252k --,--- V ?
H 3 0,
Vo< VA
? Q H
oh
8o)0
vg sin26
Vo .--- 108cm/sec;
8.104 , sec-1
[16]
17,1
Plasma waves across
magnetic field excited
by an electron beam
a n
(on. ..---- k3V0; u ..1_ ;
mg 0.)1(
(.0, ( m -r4
ao M )
o
---
4.107 sec'

[17]
v H H
, .. ,
Longitudinal electron
waves produced by an
electron beam
coH-^=k3170;
? # 7;
(08 < coir
(1)190 sin 6;
[15]
CO=y --o-- 14+ og sin2 8;
.
co_ oocose
2
2oni
oh sin0
2 (ocoH)1/2 1/1 cos 8i
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
TABLE 3. Instability Due to the Normal Doppler Effect
Form of instability
Excitation
condition.
Growth rate
F requency
Imw ,'pew
Example
I 'mu)
1 ? ?

Instability of an electron
13 e a m of oscillators
[fe=86/1 )8( v1 -vI)) 1 -
TC
ItIv5i=coff; 0--2--= ;
a klv?1 0 [here, f is the distribution function and vz = (t ? to)I i.e., the number of particles
6vz
with velocities V> vph must be greater than the number of particles with velocitiesi vz 0 can arise in the distributien function in a stellarator at
vz
the current plateau; at this point the critical field, which determines the rate at which runaway electrons are
generated, increases with time, reducing the flux of electrons through the runaway limit. The flux of runaway
electrons S is related to the distribution function f (when vz > iEs/EvTe) by the relation
OS___ e2F2 Of r eE
? , [V,-= (t ? to)]
(Here, E is the electric field and Ec is the critical electric field). A reduction in the flux of runaway electrons with
f
time leads to the appearance of a region where a > 0. The instability will occur if the growth rate 6
avz
f
z 2 WO
v is greater than the binary collision frequency. This condition can be used to determine the
6vz He
development time of the instability. It is 10-4 - 10-6 sec and agrees with the experimental data. In the present
case the frequency of the high-frequency oscillations is 1010 - 1011. This figure also agrees with the experimental
data.
3. In the analysis of instabilities it is usually assumed that the particle distribution function for the ordered
motion is Maxwellian; this assumption is then used in computing the spectra of the excited oscillations and their
growth rates.
? The important effects of low-frequency ion oscillations in a plasma in a magnetic field on diffusion processes
in the stellarator have been indicated by Spitzer [12, 31]. In our opinion, however, the diffusion coefficient given
in (311 is not convincing because the correlation function is not valid if one considers diffusion due to nonstationary
oscillations; on the other hand, the derivation is not conclusive for diffusion under stationary conditions because a
linear theory does not allow the amplitude of the stationary oscillations to be determined. The effect of beam
instabilities on the diffusion can be analyzed only for the case of a small nonlinearity.1
965
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
The distribution function can be an arbitrary function of velocity if binary collisions do not occur in a plasma.
It is important in this connection to find the necessary and sufficient conditions for an instability for the case of an
arbitrary distribution function.
The instability conditions for longitudinal oscillations in the absence of external fields have been established
by Penrose [33], Neidlinger [34], and Akhiezer, Lyubarskii and Polovin [35]. In [35] these instability criteria have
been extended to the case of a plasma in an external electric or magnetic field. If there are no external fields the
instability conditions are
f0(u) du >0; (ite)>o,
10(ue)=0; Lie
where fo (u)= f (v) dv (here, f (v) is the distribution function; v is the velocity component perpendicular
to the wave vector k and u is the velocity component parallel to k, i.e., v H ).
The conditions fo (us) = 0 and f"o(ue) > 0 mean that there is a velocity region v> ue in which lo (v) > 0.
If fo = V h) > 0, the number of particles moving faster than the wave, and thus able to feed energy to the wave,
is greater than the number of particles moving slower than the wave and capable of absorbing energy from it; this
situation leads to a growth of the oscillations in time, i.e., an instability. The condition j0t)t1 >?() implies
the possibility of propagation in the plasma of waves with phase velocities vph > ue.
The nature of an instability is important for both theoretical and practical reasons.* An instability is absolute
if an initial perturbation increases in time at any fixed point in space. However, if the initial perturbation grows
but also moves in the direction of motion of the beam, so that it diminishes in time at any given point in space,
it is a convective instability. In the case of a convective instability, a perturbation that has not had time to grow
to high values can be carried out of the system. The method of distinguishing between absolute and convective in-
stabilities given by Landau and Lifshits [36] reduces to an investigation of the asumptotic behavior (t -+ co) of the
Integral expl?ito(k)X 1] dlc that describes the wave packet.
The instability is absolute if this integral grows without limit as t co. If the integral remains finite as
t oo the instability is convective.
It has been shown by Sturrock [37] that the interaction of two opposing beams leads to an absolute instability;
on the other hand the interaction of two beams moving in the same direction leads to a convective instability. In
general, an absolute instability arises in cases where some feedback mechanism operates.
Investigations of the interaction of a beam with a hot plasma, carried out by the author together with V. I.
Kurilko and V. D. Shapiro [38], and by Sturrock [39], have shown that the instability in this case is convective.
The instability associated with a self-modulated electron-ion beam is also convective.
In a number of cases, it is difficult to establish whether a wave is attenuated or amplified because the dis-
persion equation allows a growing solution even in a system in which there are no energy sources to provide a
growing wave. For example, this is the case in the penetration of a high-frequency field into a plasma at frequencies
< wo or in the propagation of a wave in waveguides beyond cutoff. In such cases, making use of the radiation
condition we retain only the solution corresponding to attenuation. However, when a beam moves through an in-
finite system the usual radiation conditions do not hold and in order to distinguish the amplified waves we must
investigate the behavior of a perturbation in the form of a wave packet. Amplification occurs only when a per-
turbation described by a wave packet vanishes as x co for any given time t. If this requirement is not met a
growing wave can not exist in the system for x co.
A system is unstable with respect to small perturbations [proportional to exp. (? icot + Mx)], if the dispersion
equation for the system D = 0 allows complex solutions for co or k. If the frequency to is complex for some
region in which k is real the instability can be absolute or convective.
966
[ Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
It has been shown by V. I. Kurilko [40] for example, that when a plasma moi/es through a plasma waveguide,
the complex values of k correspond to evanescence rather than amplification. If a plasma moves in a medium with
a dielectric constant a in crossed electric and magnetic fields (V0 > ? or Vo <
E
it is found that real
values of co correspond to complex values of k. However, amplification occurs only when Vo >) while Vo <
E
corresponds to evanescence [41].
E
Criteria for distinguishing between gain and evanescence and a simple method for distinguishing between
absolute and convective instabilities have been given by Sturrock [37]. A comprehensive discussion of these criteria
and their limits of applicability has been given by R. V. Polovin [41].
4. Above, we have considered the interaction of infinite beams and plasmas. For this reason, strictly speaking
the relations we have obtained apply only when the geometrical dimensions of the system L11 and Li, are much
greater than a wavelength, that is to say
,11L11>>1, x_LL I.
These conditions are not satisfied in many systems and for this reason there is a considerable discrepancy
between the theoretical and experimental results. The most striking difference between finite and infinite plasmas
Is found in the absence of a magnetic field. Slow transverse waves cannot propagate in an infinite plasma in the
absence of a magnetic field because E = 1 < 1 at all frequencies. Obviously, under these conditions there
cannot be an effective interaction between free particles and waves in the plasma; in particular, the Cerenkov con-
dition cannot be satisfied. ?
If the plasma is bounded in the radial direction, slow waves can propagate in spite of the fact that can be
smaller than unity and even negative in the region occupied by the plasma. A similar situation obtains in a plasma
bounded in the direction of propagation of the wave, in particular, a spatially perioClic plasma. In this case the
system is again capable of propagating slow waves even when the dielectric constant is smaller than unity or nega-
tive; in addition, a system of this kind exhibits the properties of an anisotropic medium.
The dispersion properties of a bounded plasma in a magnetic field determined by the anisotropy and gyro?
-
magnetic effects of the medium must be considered in conjunction with the waveguide properties associated with
the geometry of the system. This again leads to a marked contrast in the dispersion ;properties of finite and infinite
plasmas. Inasmuch as the elementary processes responsible for the interaction of the beam with the plasma are
sensitive to the dispersion of the system, i.e., the dependence of phase velocity on 6equency, one expects that the
instability criteria will be modified in going from an infinite plasma to a finite plasma (in particular, the critical
velocity and the beam current); one expects that the frequency spectra and growth rates would also be modified.
Investigations of instabilities in finite beam-plasma systems have shown that if the wavelength in the plasma
(V0/w0) is comparable with the beam radius a then the wavelength corresponding to the onset of instabilities be-
comes greater and the growth rate becomes smaller. For example, in the electron-ion beams which have been
considered by G. I. Budker [42], the ratio of growth rates for a finite plasma and an infinite plasma is
iMO) (a < X) V0 e 40,2
mo) (a A,) cooa
woa 1.
Vo
In the excitation of longitudinal plasma waves by a beam, a situation which has been considered by M. F.
Gorbatenko [17] and by Sturrock [39] and in [43], the reduction in growth rate can be quite large ? a factor of
5 or 10. The amplitude of the waves is thus reduced markedly. The parameters can be chosen in such a way that
the minimum wavelength is greater than the length of the system. Under these conditions the instabilities in
? According to Neufeld and Doyle [77], when a beam interacts with a plasma it is possible to excite electromagnetic
waves under these conditions; In contrast with the case of an individual particle moving through a plasma, the
presence of a beam in the plasma changes the dielectric constant and can reduce the phase velocity to values equal
to the velocities of particles in the beam.
967
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
question will not be excited. This feature of the interaction of finite beams with a plasma is important in connection
with the search for methods for suppressing plasma instabilities.
5. In spite of the successes which have been achieved in the theory of particle-plasma interactions, the dis-
covery of various new instabilities, and the experimental verification of the basic theory, our level of understanding
in this field is still not sufficient to allow evaluation of these instabilities in processes occurring in various kinds of
discharges. This situation is a result of the fact that most of the analyses that have been carried out can only deter-
mine the spectrum of excited oscillations and the growth rates. These analyses do not allow us to determine one of
the most important characteristics ? the oscillation amplitudes. To determine these amplitudes we must know the
energy spectrum of the initial fluctuations and must take account of the feedback effect of the oscillations on the
plasma distribution function and the motion of the beam particles; the transition from linear to nonlinear regimes
must also be considered and the amplitude of the stationary oscillations must be known.
It should be noted that nonlinear effects become important at relatively low electric fields. For nonresonant
eE02t.
plasma waves the criterion for linearity is 2 < 1,. Hence, nonlinear effects are very important
p
no h
( i _ _IV ) V h
P
at long wavelengths and small phase velocities. In most cases the excitation conditions reduce to the requirement
Vo >r, VD', , so that the role of nonlinear effects in the presence of beams becomes still more important. It can be
shown that in the case of ion oscillations nonlinear effects appear at very high field intensities. Actually, however,
nonlinear effects should appear even at small intensities since the ion interaction is strongest with slow waves and,
at the low frequencies corresponding to ion oscillations,slow waves imply long wavelengths X. For example, with
8 ph 10-3, we have X 103 and Ecr < 10 v/cm. Nonlinear effects are almost certain to appear at resonances.
wH ?
For example, in the presence of a magnetic field the criterion for nonlinearity contains the factor (i )-1
ca '
As the amplitude of the excited wave increases,a number of nonlinear effects appear that tend to weaken the
Instability [45-48]. Since the phase velocity of the wave depends on its amplitude, as the amplitude increases the
synchronism between the beam and the wave is disturbed and the effectiveness of the interaction is reduced. Syn-
chronism can also be distributed because of a reduction in beam velocity resulting from the loss of energy in the
excitation of oscillations. The feedback effect of the excited oscillations on the distribution function for the
electrons in the beam means that the number of particles giving energy to the wave is reduced while the number
of particles absorbing energy from the wave is increased. Thus, the distribution function is equalized in the region
V increase h and the instability is suppressed [48, 45, 46]. Nonlinear effects in a beam ? plasma interaction also tend
to the temperature of the beam and the plasma even when there are no binary collisions.
VT
_I_ 6
The first and third mechanisms are important when ? ?; the second mechanism is important when
Vo w
VT 6
?
Vo
These mechanisms are listed in Table 4 for purposes of illustration.
Examination of this table indicates that,in general,nonlinear effects tend to do the following: 1) reduce
growth rates, 2) limit the ordered velocity of the beam and the reduction in conductivity a (a = I /E) of the
plasma due to the conversion of energy of ordered motion into oscillation energy, and 3) increase beam tem-
perature.
These conclusions apply for small nonlinearities only; it is still necessary to develop a theory for arbitrary
nonlinearities. The existence condition for arbitrary nonlinear periodic solutions for the interaction in question has
been given in [50]. At the present time the problem of beam interaction with a plasma for an arbitrary nonlinearity
has been treated only for the excitation of stationary electron and ion longitudinal oscillations. This analysis has
made it possible to determine the maximum amplitude of the field, taking account of the thermal motion and the
maximum field gradients which are proportional to the variation of plasma density (see Table 5).
? These nonlinear effects has been observed experimentally by 0. G. Zagorodnov, Ya. B. Fainberg, B. I. Ivanov
and L. I. Bolotin [44].
968
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
TABLE 4. Nonlinear Effects in Beam-Plasma Interactions
Type of
interaction
Retardauon of the
beam
(AV)
Temp. increase
(13 91)
Relaxation time
. (Trel)
Increment
v 0
P
Lit.
reference
Electron beam in ion
Electron
plasma
(Eo > Ecr; Of =m;
Eo=eno2/3; k2DED=e;
?-=(00t)
Vo III (QE -Y
u_L ,E.,3 J
e"

0 _,,o(Eq)3 e" .
U --c-, -2/
I
--A
?
Vou1/3 2? 4/8
27
X (1 -1-1-6
21/3e2aF*
[23]; [45]
H t
i. ? e,0 T , 2
EDe"
0 '=u.
in C-j
a0)5 .0
E3
x ?q a_5/2 )
x
at5)2 ; a=0,05
)1/2
E ? 'o ( 3/2
o
Y
- M2V804141/3
Monoenergetic beam
In a plasma
V vo
?
w1/20 cood0
_. ,2 0 0
011
1 In
0)06
mvg
x (00
e2E4F --\
vo
5 e2EZF
[46]; [47]
?
15%11'
' 17.2 %, eto '
-)X
(vri g = ( ) ;
z
--F,dmVO
no v 1-oT5 5 3 1
g. 2)
6?)? (1 ?863rn2V34 )
X 1 -i-6+1 -.0
( T
8 6 3m2V84
-
442
Ionacoustic oscillations
(monochromatic wave)
_
_
A e2E4 2SOhdT
?
Oh (t) ?
[18]; [45]
exp
Ott (0)
e2Ek2k2
X m2
1
x 4,d exp
0 vk
'2
0
'i ? MO e
High-frequency
-oscillations of an
electron beam with a
plasma
(6) --'-'??)H-FRIIV?)
AO _L
A01 3 1/. (0,11
?
_
_
[49]
=
Oo It %VS -
1 5
x 7 exp [ -T
( coo nt "ro .r 1
mV0
\-(01/ no I
t }
? F?exp 2 S oh ds
1 o
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Literature
reference
-
7
c\i"
..o
(iiot?Tu !8ol--01t)
aIduirxa
C
!I
...
S
a
? " -
- 0
11 Ti
0
0
n
4::
---.
a
t4D
a.
I E
c.)
0
"
-,
il
X
Ctt
E
'414
r----\
L.,
I
...3?.
(------,
,?_-)
tt)
-,t
4
CV
g
8
8
cd
vi
I
g
7-1
E,
t
I ?CL.
I
I g
c\I
-.'
.4
f:l?
g
-o-
1--",
IM
Z'r I
---,
'a.'Icl'
oo
I
...
tr
..I,
---J
X
Type of
interaction
Electron oscillations in a
plasma
Electron oscillations in a
plasma with a beam
Ion oscillations
Examination of Table 5 indicates that the ampli-
tudes of the stationary oscillations can be very high.
For example, at a plasma density n 10M and Vph
108, we find Emax 30 kv/cm. There is a maximum
field for longitudinal waves above which the solutions
are no longer unique. Since shock waves cannot appear
in the one-dimensional case (if ion motion is not taken
into account), the nonunique solutions would appear to
correspond to opposing particle streams. It also follows
from Table 5 that as the number of streams increases,
the field intensity at which beam breakup occurs is
reduced. These effects are of great importance in the
development of new methods of plasma thermalization
and the realization of CTR with opposing particle
streams.
An important question is the stability of high-
amplitude longitudinal waves that are excited by beams.
Recent studies of high-amplitude longitudinal waves
excited by a monoenergetic beam [54, 55] show that
these waves are unstable, at least, against perturbations
with wavelengths appreciably smaller than the wave-
length of the stationary oscillations. The nonlinear
solutions in the form of bounded pulses excited in the
interaction of a beam with a plasma are also unstable.
Since the analysis of nonlinear problems is a problem
of great mathematical difficulty, the study of transitions
from linear regimes to nonlinear regimes, the effects of
spectral decay,* and the formation of opposing beams
and breakup of the particle streams are most conveniently
carried out by means of high-speed computing machines.
An investigation of the formation of opposing beams and
the thermalization of plasma at high oscillation ampli-
tudes [23, 56] shows that the strong interaction
between individual plasma layers produces a relative
motion which, in turn, converts the initial wave energy
into energy of relative motion of the individual layers.
The relaxation time for this process is very short. For
example, [56], if the initial amplitude of the oscillations
is 7910 greater than the critical amplitude, 70-80% of
the wave energy is transformed into energy of relative
motion within 10 Langmuir periods. If a relatively
small disordered motion is superposed on the initial
oscillation (additional energy amounting to about 5%
of the initial wave energy) the relaxation time is re-
duced to a single period of the high-frequency oscilla-
tions. Thus, the mechanism under discussion is very
effective for transforming the energy of ordered os-
cillatory motion into energy of relative motion of
plasma particles.
? Spectral-decay is the name given to the nonlinear interaction of waves with nonparallel wave vectors, which
leads to the generation of new waves (this process is accompanied by the dissipation of the energy of the initial
waves in a time of the order of 10-8 sec [76]).
970
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
The results cited above refer to electron oscillations. It would appear that there are conditions for which
similar processes would obtain for ion oscillations. In this case, the important thing would be the long-lived relative
motion of the ions, a feature of interest in the development of methods for realizing CTR.
6. Before analyzing the available experimental data we shall dwell briefly on possible methods of suppressing
instabilities; this is the most important aspect of the problem. Work on this problem has only started, so that the
considerations given below are necessarily qualitative. In the final analysis, instabilities arise as a result of one or
more basic interaction mechanisms together with phase focusing or particle bunching, the latter causing the inter-
action to be enhanced because of coherence. Thus, to avoid instabilities we try to produce conditions for which
these basic mechanisms cannot operate and for which phase focusing or bunching of the particles cannot occur.
If the Cerenkov effect and the anomalous Doppler effect are to operate,the velocity of the particle or
oscillator must be greater than or equal to the phase velocity of the wave. Hence, an instability can be avoided
If this condition is disturbed by changing the velocity of the wave or the velocity of the particle, changes in the
dispersion properties, in particular the phase velocity of a wave, can be achieved byl varying the system geometry
(for finite beams and plasma), by placing the plasma in a magnetic shell, by utilizing the dependence of phase
velocity on amplitude (this effect can automatically limit the instability because ah increase in amplitude causes
a change in phase velocity and thereby disturbs the synchronism between the field and the wave), and by changing
the relative concentration of various components in a multi-component (for example, DT or DH) plasma.
By changing the phase velocity of the wave we can completely suppress the instability or, at least, modify
Its frequency spectrum and reduce the growth rate. By changing the velocity of the beam particles we can disturb
the condition V ? VP h and completely avoid the instability or modify the oscillation spectrum markedly. Ion
oscillations are excited if the particle velocity lies in the range VT4L1/2 < Vo < VTC. If the particle velocity
V0> VTe , then electron oscillations are excited, but ion oscillations are not. The beam velocity may change
automatically as it is retarded. Furthermore, there may be a rapid increase in beam velocity during the relaxation
time T rel. during which corresponding instabilities cannot develop. From this point of view the stellarator is at an
Important disadvantage because the electron velocities in this machine vary slowly from v = 0 to vmax and under
these conditions a succession of instabilities may be encountered. Injection of a plasma in the stellarator at some
finite velocity would reduce the spectrum of instabilities and would facilitate the suppression of instabilities.
The conditions for the basic mechanisms such as the Cerenkov effect and the Doppler effect can also be
disturbed as a result of nonlinear processes associated with the motion of the particles and oscillators,and nonlinear
effects associated with the propagation of waves in the plasma. As is well-known [571, the basic mechanisms re-
sponsible for instabilities can be interpreted in terms of a resonance between the characteristic oscillations of the
system (plasma in the present case) and a driving force, which is associated with the motion of the charged particles
and oscillators., In this interpretation the Cerenkov condition and the Doppler effect become resonance relations.
The nonlinear effects indicated above disturb the resonance relationships and thus serve to inhibit instabilities. On
the other hand, it is possible that other nonlinear resonances can arise; these require further investigation.
Another way to avoid instabilities is to prevent the phase focusing (particle bunching) process which is re-
sponsible for the coherent interactions.
The following techniques may be capable of weakening instabilities or even suppressing them entirely:
1) Introducing an artificial spread in beam velocity. For purposes of illustration we compare the growth rate
6(VTe = 0) when a plasma is excited by a monoenergetic beam and when it is excited by a beam with a weak
(VTe < V0) thermal spread 6:
V1/3 V 2
(ii. 1/8; r ( ni 0 /r7
Te
no)no I , VTe
It is evident that a velocity spread tends to reduce the growth rate considerably or can even remove it com-
pletely:*
2) Premodulation of the beam. It is well-known that the exponential growth of an instability is due to the
fact that the field caused by the onset of the instability increases the particle bunching and that the enhanced
? It should be noted, however, that in a number of cases a large thermal spread in the beam velocity can lead to
the production of new instabilities; in general however, these are characterized by small growth rates.
971
I Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP110-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
bunching leads to further amplification of the field. Premodulation of a beam at some desired frequency serves to
inhibit the bunching at other frequencies and can thus serve to suppress a whole spectrum of instabilities. In this
case we must make sure that the coherence conditions a < X I are not satisfied for characteristic plasma oscilla-
tions. Premodulation of the beam may make it possible to suppress an instability at some given frequency. This
result can be accomplished if the wavelength of the modulation X m is made x m = a' /2. It should be noted that
in avoiding ordinary instabilities we may produce instabilities associated with parametric resonances. However, the
bandwidths of parametric resonances are small so that under actual conditions the effects of inhomogeneities or
collisions can suppress these instabilities;
3) Variable system geometry. As we have indicated above, the frequency spectra and growth rates are com-
pletely different for finite and infinite plasmas. By changing the geometry of a system we can produce conditions
for which the minimum wavelength of an instability becomes greater than the dimensions of the system, so that the
instability cannot be supported;
4) Feedback effect of the excited oscillations on the distribution function causing the number of particles
giving energy to the wave to become equal to the number of particles absorbing energy from the wave;
5) Exploiting the nonlinear nature of !he motion of particles and oscillators;
av h
6) Varying the phase velocity along the system in such a way that az P.. < 0. In actual systems this con-
dition can be easily realized by changing the plasma parameters, for example the density. In a number of systems
it can be satisfied automatically.
Comparison of the Theoretical and Experimental Results
'7. Because of the complexity of the processes involved in the interaction of charged particles with a plasma
and the importance of nonlinear effects, the experimental investigations in this field are of extreme importance.
The basic problems of the experimenter are to discover instabilities, to determine the conditions under which
they arise, to determine their frequency spectra, growth rates, and relaxation lengths, to measure the loss of energy
of ordered electron motion in the excitation of oscillations, and to determine the effect of these instabilities on
plasma conductivity and diffusion. This information is then used in the important problem of devising techniques
for suppressing instabilities.
As is well known, experimental research on plasma oscillations started with the work of Langmuir and Penning
in 1921-1929 [58]. An investigation of oscillation processes in plasma carried out in 1939 by Merrill and Webb [59]
was of great value. The instabilities due to excitation of longitudinal plasma waves by a mean of electrons in a
uniform plasma with no magnetic field were demonstrated experimentally in 1957-1958. The basic characteristics
of the instabilities were determined in this work.
We now describe some of the general results obtained in this work.
In an experiment carried out by Kharchenko, Nikolaev, Lutsenko and Padenko and the author [60] it was
shown that an initially unmodulated electron beam passing through a plasma (with no magnetic field) excites
longitudinal high-frequency oscillations at frequencies close to the Langmuir frequency. The excitation of these
oscillations is accompanied by self-modulation of the beam. The energy loss per beam particle was approximately
100 ev/cm; this is to be compared with the energy loss of a single charged particle for this case? approximately
10-6 ev/cm. The high losses are due to the coherent nature of the interaction between the beam and the plasma
that results from the self-modulation caused by plasma oscillations. In these experiments, the effect of plasma in-
homogeneities, in particular the effect of sheaths in the region of the cathode fall, was avoided by using a high-
frequency discharge and injecting an electron beam from outside at an energy of 50-70 key, an energy appreciably
greater than the energy acquired by an electron in the boundary Debye layer. The initial perturbations were due
to fluctuations in the plasma and in the beam.
The quantity a is the length of a particle bunch.
972
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Papers by Demirkhanov, Gevorkov, Popov, et al. 162] describe the observation of longitudinal oscillations,
the determination of the relaxation length, and the production of harmonics of the excited waves.
The spatial structure of the excitation region and the energy lost by a beam in moving through the plasma
have been studied by Gabovich and Pasechnik [63].
The amplification of electromagnetic waves due to interaction of a beam with longitudinal waves in a plasma
has been observed in experiments by Boyd, Field, and Gould and by Kislov, Bogdanov, J,and Chernov [65,66J. In
these experiments the initial perturbation was imposed by an external source and growing high- frequency longitudinal
waves were excited in the system.*
These experiments were found to be in contradiction to the experiments carried out by Looney and Brown [67]
In which beam instabilities were not observed. The reason for this discrepancy is the fact that Looney and Brown
used beams with small transverse dimensions. We have indicated earlier that the growth' rate is diminished when
.1
the beam radius is reduced. The theoretical growth rate for the Looney and Brown experiments is approximately
2.5 cm-I. Hence the field should increase by one order of magnitude for a beam-plasipa interaction length of
1.5 cm. In these experiments the initial perturbation was due only to fluctuations in t4e beam and plasma so that
the oscillation amplitudes were small and difficult to detect. In contrast with the Looney-Brown experiments, in
all the experiments listed above the beam radius and interaction length were quite larger. The theoretical gain in
these experiments is approximately 103-105.
The theoretical and experimental growth rates are found to be in satisfactory agement (Table 6). There
are some discrepancies in the experimental data obtained by different authors and between the experimental data
and the theoretical values for the relaxation length for the beam-plasma interaction, iJe., the distance in which
the energy of the ordered motion is converted into oscillation energy. Before a more meaningful comparison of the
experimental and theoretical data can be made it will be necessary to refine the measurements of beam velocity
spread and plasma temperatures because these parameters have an important effect on: relaxation length.
Experiments on the interaction of charged-particle beams and a plasma in a magnetic field were carried out
in 1959-1960 (I. F. Kharchenko et al.) [68]. In these experiments both low-frequency and high-frequency oscilla-
tions were excited and both modulated and unmodulated beams were used. The high- frequency spectrum is de-
scribed well by the theoretical relation co 17'co:-F cofi; (oil. These oscillation's are especially strong if
coo col/ for the case co . The low-frequency oscillations are distributed in bands at tens, hundreds
and thousands of kilocycles. These occur near regions corresponding to ion-cyclotron and magnetohydrodynamic
waves. In order to obtain more accurate information on the kinds of waves that are excited it will be necessary to
measure the wavelength (8 h) and to determine the angular dependence of the electromagnetic fields.
Because the oscillation frequencies are low, in investigations of beam-plasma interactions under pulsed con-
ditions one can observe the development of an instability and determine the growth time. Oscillations at frequencies
of 100 kc are characterized by an instability development time of approximately 50 ?sec, i.e., approximately ten
oscillation periods [49].
In addition to the observations of low-frequency and high-frequency oscillations cited above, there are ob-
servations of parametric oscillations in a modulated beam.
Examination of Table 6 reveals the following:
1) At the present time most of the instabilities analyzed above have been observed experimentally;
2) The basic features of the instabilities (excitation conditions, growth rates, frequency spectra, relaxation
lengths) have been shown experimentally to be in agreement with the theoretical results in most cases;
3) The development of an instability is accompanied by oscillations and a marked reduction in the energy
of ordered motion of the beam. Even in a rarified plasma (no 1011) the losses are approximately 100 evicm per
particle;
* The growth rate in space was determined from the ratio of powers at the output and input of the system.
973
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
TABLE 6. Comparison of Theoretical and Experimental Values of Growth Rates and Relaxation Lengths.
Form of
oscillation
Frequency
Growth rate
Relaxation length
experiment
theory
Amplification of
? longitudinal waves
in E beam by a
plasma
?0o
(2.1010)
experiment theory
Lit.
ref.
[64]
Excitation of longi-
tudinal waves in a
plasma by a beam
100
(2.1010)
0.16coo
fm 0.15coo
dS eV
?50?
dx cm
cl S == 100 eV
dx cm
[601
Amplification of
longitudinal waves
in abeam by plasma
Excitation of longi-
tudinal waves in a
plasma by a beam
? 010
(2.1010)
Imk 0.2 cm-'
Imk 0.15 cm-1
[661
? 100
(1.5.101?)
Imk 0.12 cm'
fmk -= 0.15 'cm-1
7 cm
8.4 cm
(61]
Scattering of a
beam on plasma
oscillations
3.109 to 3.1010
1 cm
4-10 cm
[631
High- frequency
osallations in a
magnetic field
coH (1010);
(,)
(2 to4.1010)
1 to2.108;
1 to2?109
0.5.109
[68]
Low-frequency
oscillations in a
magnetic field
? (6 to 12O)? 103;
? DB- (6 to.30)?105
1 to20.103
(nO"In
orel 5.10-8
[68]
High- frequency
oscillations in a
magnetic field
H (4.2.109);
(04+4)1/2
? 108
[69]
Low- frequency
oscillations in a
magnetic field
Low-frequency ion
oscillations
(oil< co< (4+(01)112
Imk 0,7
01=00
Imk 0.7
(10 =00
(701
6.105
(?o)
[711
High-frequency
oscillations in a
magnetic field
1 to2.109
[72]*
* The author is indebted to S. D. Vinter for acquainting him with this work.
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Dec assified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
4) The development of an instability also leads to a marked increase in the energy of the plasma electrons.
For example, in experiments on beam-plasma interactions in a magnetic field this energy is 10-20 key [68];
5) The development of an instability is accompanied by an appreciable increase in the ion and electron flow
across the magnetic field.
Thus, all the predicted "dangerous" consequences of instabilities have been established experimentally.
In addition, however, experiments that have been carried out indicate the possibility of suppressing instabilities.
For example, beam modulation can serve to inhibit a whole spectrum of instabilities because the modulated beam
can excite oscillations at wavelengths far from the resonance wavelength. On the other hand, modulation of beams
and plasma can cause parametric instabilities, but since the width of the excitation region is very small, the pre-
sence of a density inhomogeneity in the plasma serves to suppress these parametric instabilities. A number of in-
stabilities are also suppressed by a relatively small spread in beam velocity. In general, the growth rate is a sen-
sitive function of system geometry, i.e., the radius of the beam and the plasma; hence, an instability can also be
suppressed by choosing appropriate values for these parameters. Finally, a number of experiments indicate that the
Increasing amplitude of the excited waves leads to nonlinear effects which then limit the development of instability.
Further research into instabilities and techniques for suppressing instabilities will require experiments in which
the plasma density can be increased to no..-,, 1013 - 1014 and in which the degree of ionization can be made 50-100%.
It will also be necessary to increase the electron current to I-_,.? 102 - 103 amp. These experiments will probably be
carried out most conveniently in plasmas produced by powerful high-frequency discharges, in a cesium plasma, or
In a plasma formed in a linear betatron. It will also be necessary to carry out experiments in which direct observa-
tions can be made of the effect of oscillations on conductivity and diffusion in a high-density plasma.
We have been speaking of experiments in which beam-plasma interactions were investigated in systems de-
signed especially for this purpose, in which the interaction was observed in 'pure" form, uncomplicated by other
processes.
1
There are experiments in which beam-plasma interactions have been studied directly in systems designed for
investigating CTR; among these we may mention the work of Bernstein et al. [731 and Ellis et al. [74]. The most
important result of this work is the experimental proof that instabilities arise in the. stellerator at currents appreciably
below the critical current associated with magnetohydrodynamic instabilities. The observed instabilities are
accompanied by an increase in the high-frequency noise level, which is appreciably above the thermal noise level.
Radiation is observed in the frequency region 10,000-70.000 Mc corresponding to Langmuir oscillations. Measure-
ments of the x-ray radiation show that the instability is accompanied by the appearance of electrons with high
energies ? up to 3 Mev. The work by Ellis et al. [741 showed that the instability develops when conditions are
satisfied for the generation of an appreciable number of runaway electrons. In this case, the energy acquired by an
( E
electron in one mean free path must be comparable with the energy of the thermal motion y = ---, c 1 . In
ter
these experiments the quantity y is 0.1-0.2. When these values are reached the current in a stellarator falls off
sharply in spite of the increasing electric field. This reduced current can be explained by assuming that the elec-
trons lose energy in the excitation of oscillations. The time in which y reaches the critical value and in which
the sharp current reduction is observed is approximately 200-800 ? sec. The experiments considered here indicate
the important role of high-frequency instabilities in the stellarator. It is as yet difficult, however, to evaluate the
effect of high-frequency oscillations on plasma diffusion in the stellarator.* A shortcoming of the experiments is
the fact that the spectrum of excited oscillations was not investigated and that no correlation was looked for between
these oscillations and the conditions for onset of an instability. Among other things, the maximum of the excited
frequencies is related simply to the number of runaway electrons. These frequencies are (c000-Hoiii)1/3X mio/o3,
where woi is the ion Langmuir frequency, we is the plasma Langmuir frequency for the plasma at rest and woe is
the Langmuir frequency of the moving plasma.
8. At the present time the main effort in beam-plasma research has been concentrated on instabilities, since
these represent one of the chief stumbling blocks in attempts to achieve CTR. It should be noted, however, that the
* The low-frequency instabilities should have the greatest effect on diffusion in the stellarator.
975
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
beam-plasma interaction also provides a strong "collisionless" exchange of energy between the charged particles and
the plasma and this can play a useful role. It is of both theoretical and experimental interest to investigate the use
of this mode of energy exchange for plasma thermalization. As we have indicated above, when electron beams
interact with a plasma the energy of the ordered electron motion can be converted into longitudinal- wave energy
in a very short time (approximately 10-6 - 10-8 sec). As the amplitude of these waves increases, they can cause
relative motion of electrons and ions with appreciable velocities. Relative motion also appears in instabilities due
to transverse motion, for example cyclotron resonances.
An effective transfer of energy from the electron beam to the plasma ions is possible in this interaction. The
dimensions of the electron bunches or the modulation wavelength a must be such that the conditions for coherent
interactions are satisfied for the ion oscillations but not for the electron oscillations. It is necessary that xe< a
> 14 Mev, only data on U238 have been published up to now [17]. Since monoenergetic neutrons with variable
energies greater than 20 Mev are almost impossible to produce, spectral analysis of neutrons with respect to the
994
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
E (1=11. 2
13.6'
15.5
1Z5
19.6' Mev
1" = 1.10
1.16'
1.25
437
1.48 Mev
1
^.
cl. N
LC Zsi
1
0 2 6' 8 10 ! 12 E*, Mev
1
Fig. 9. Density of levels for the neutron spectrum of the V (a, n) reaction Or a 900 angle.
200
100
011,
? 0,
Ic
Il
t9'
0
?
a
20 40
CO 80 100
120 4. mi:Isec
Fig. 10. Time distribution of pulses due to Th 232 fission by neutronsi, from
the D (d, n) reaction for Ed =19.5 Mev. The range was 3 m. A) Fission
by neutrons from the D (d, n) Hes reaction; B) fission by neutrons from
the D(d, pn) D reaction.
995
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
flight time should be used. However, this problem requires the development of fast-acting detectors capable of
clearly distinguishing fission events from a-decay events. The available solid scintillators are not suitable for this
purpose due to the nonlinearity of the luminescence yield. Therefore, gaseous scintillators were developed and
utilized [18]. The connection of these scintillators to the circuit of the spectrometer based on flight time made it
possible to observe the entire complex spectrum of neutrons that cause fission. Figure 10 shows the time distribution
of pulses due to Th232 fission, which is caused by neutrons from the D(d, n) reaction at Ed = 20 Mev. The broad
group corresponds to breakup neutrons, while the narrow group (two peaks) corresponds to neutrons from the D(d, OHO
reaction. Similar results were also obtained for neutrons from the T (d, n) reaction, where the relative intensity of
the continuous spectrum is even greater.
Figure 11 shows the dependence of the Th232 fission cross section on neutron energy in the 3-37 Mev interval.
6, b
1.0
0.9
0.8
0.7
0,5
0.4
0.3
0,2
0.1
5 10 15 20 25 30 35
En, Mev
Fig. 11. Cross section of Th232 fission by neutrons with energies from 3 to 37
Mev. 0) With T(p, n) Hes neutrons; 0) with D (d, n) Hes neutrons; x)
with T (d, n) Hes neutrons.
Investigation of Nuclear Reactions That Are Accompanied by the Formation
of Tritium
Tritium is one of the usual products of the very simple (p. t), (d, t), (a, t), etc. nuclear reactions. The
investigation of these reactions is of interest for an explanation of their mechanism and also from the point of view
of nuclear spectrometry. In connection with this, a simple but sufficiently efficient method for studying the
spectra and angular distributions of tritons has been developed in our laboratory [19]. Piles of aluminum foils,
which are penetrated by tritons to depths that depend on their energy, were arranged at different angles around the
thin target, which was bombarded by accelerated particles (Fig. 12). After exposure to the beam, tritium was
extracted by heating from each foil and was introduced into a Geiger counter. The amount of accumulated
tritium was determined with respect to the measured activity, while its energy was measured with respect to
the depth at which the foil was located. A single irradiation with a flux of the order of 100 ? amp ?hr made it
possible to measure triton spectra in a wide energy interval (from 0 to 20-30 Mev) and also the angular distributions
in an interval from 70 to 1500.
In spite of its relative simplicity this method is very efficient, since the recording of any other particles,
except those of tritium, is precluded, and therefore, the background that limits the possibilities of other methods
is absent.
996
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
IV
30
25
20
15
10
5
Fig. 12. Device for studying nuclear reactions in which tritium is
generated. 1) Target; 2) entrance collimator; 3) holders with
collecting foils; 4) Faraday cylinder; 5) magnet which prevents
the escape of secondary electrons; 6) chambers for targets which
are oxidizable in air.
0
2.18
43770
&535g
53
q6:63
?
' ? ?
5 10 15 20 E, Mev
Figure 13 shows the spectrum of tritons from the
Li7 (d. 0 Le reaction, which was observed under a 7? angle.
The monoenergetic lines near the upper limit of the spectrum
correspond to the lower level of the final Li8 nucleus. Their
intensity sharply decreases with an increase in the Li8 ex-
citation energy, which is typical for the pick-up reaction
(d, t). The angular distribution of the three groups corresponds
to the extraction of neutrons with the orbital moment 1 =1.
The continuous triton spectrum is connected with multiparticle
reactions.
The direct process of neutron extraction in the (d, 0
reaction primarily leads to the excitation of hole states of
the final nucleus, and it makes it possible to determine the
binding energy corresponding to the single-particle state of
the neutron in the target-nucleus. Thus, for instance, the
investigation of the 018 (d, t) 017 and F19 (d, 0 F18 reactions
shows that the energies of the s and d states are interchanged
in transition from 018 to F19 [20].
The possibility of knocking out deuterons from filled in-
ternal nuclear shells is very interesting. The effect of various
shells is most clearly manifested in (d, 0 reactions with zir-
conium isotopes [21]. The spectra of tritons from these
Fig. 13. Spectrum of tritons from the , ) e
reactions are shown in Fig. 14. Two groups of tritons can be
Li7 (d t L
observed in these spectra. The intensity of one of them is
reaction for a 70 angle at Ed = 20 Mev.
proportional to the number of supermagnetic neutrons. It is
obvious that this group is formed by the knocking out of the external neutrons. The intensity of the second group
Is practically equal for all isotopes, including the magic Zr99, the spectrum of which is not shown here. This group
is connected with the extraction of neutrons from filled shells. The angular distributions of both groups are matched
997
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
with the neutron orbital moments, which are assigned to the groups by the well-known shell scheme (1 = 2 for the
first group, and 1 = 1 and 1 = 4 for the second group).
14
12
10
8
2
Zr91(d,t)Zr"
xc
Mev 4 0 Mev 2
?exc , exc Mev
P
f, Me 5 3 1,0 En, Mev 7 5 4 3 2 En, Mev 7 5 3 2
L 1
Fig. 14. Spectrum of tritons from the (d, t) reaction with zirconium isotopes.
In 018 (d, t) 017 and F18 (d, t) F18 reactions, the extraction of neutrons from closed p-shells was also observed
[20]. At the present time, the problem of the extraction of nucleons from closed shells is being solved by using the
method of the quasielastic scattering of fast protons on nuclear protons [22]. In these experiments, it was also
possible to separate the effects of various shells.
Besides the (d, t) reaction, other nuclear reactions were also studied by means of this method [23]. An in-
vestigation of the Nell ( a, t) reaction showed that its mechanism sharply differs from the mechanism of the Na23
(d, n) reaction, although the transfer of protons to target-nuclei takes place in both reactions.
The above projects were completed by the large team of co- workers of the cyclotron laboratory with the
participation of the authors of the present article.
We must mention also the large contribution made by N. Venikov who, since 1957, conducted work on the
improvement and utilization of the cyclotron, and the important\role played by M. A. Egorov and N. N. Khaldin,
who supervised the preparation of the experimental equipment.
LITERATURE CITED
1. L. M. Nemenov et al., Atomnaya Energiya 2, 1, 36 (1957).
2. A. V. Antonov et al., Pribory i Tekhnika floperimenta 1, 41 (1958); A. V. Antonov et al., Pribory i
Tekhnika iksperimenta 6, 20 (1959); B. S. Kazachina et al., Pribory i Tekhnika ifksperimenta 1, 110
(1960).
3. N. A. Vlasov et al., Zhur. Eksperim. i Teoret. Fiz. 28, 639 (1955).
4. L. F. Kondrashev et al., Pribory i Tekhnika Eksperimenta. 1, 17 (1958).
5. K. P. Artemov, N. A. Vlasov and L. N. Samoilov, Zhur. Eksperim. i Teoret. Fiz. 37, 1183 (1259).
6. G. F. Bogdanov et al., Atomnaya Energiya 1, 66 (1956); G. F. Bogdanov et al., Atomnaya Energiya 3, 9,
204 (1957).
7. A. A. Kurashov et al., Atomnaya Energiya 5, 2, 135 (1958); B. V. Rybakov and V. A. Sidorov, Fast-Neutron
Spectrometry [in Russian] (Atomizdat, Moscow, 1958).
8. G. P. Mel'nikov, L. I. Artemenkov and Yu. M. Golubev, Pribory i Tekhnika Eksperimenta 6, 57 (1957).
9. G. F. Bogdanov et al., Zhur. Eksperim. i Teoret. Fiz. 30, 981 (1956).
10. G. F. Bogdanov et al., Zhur. Eksperim i Teoret. Fiz. 30, 981 (1956); G. F. Bogdanov et al., Zhur.
Eksperim. I Teoret. Fiz. 30, 185 (1956); G. F. Bogdanov, N. A. Vlasov, S. P. Kalinin, B. V. Rybakov and
V. A. Sidorov, Transactions of the Columbia Conference (1957).
11. L. Granberg, A. Armstrong and R. Henkel, Phys. Rev. 104, 1639 (1956); J. Brolliy et al. Phys. Rev. 109, 1277
(1958).
998
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
12. N. A. Vlasov et al., Transactions of the Paris Conference (1958); N. A. Vlasovr et al., Zhur. Eksperim. i
Teoret. Fiz. 38, 1733 (1960).
13. B. W. Rybakov, W. A. Sidorov and N. A. Vlasov, Nucl. Physics 23, 491 (1961)1.
14. K. P. Artemov and N. A. Vlasov, Zhur. Eksperim. i Teoret. Fiz. 39, 1612 (1960).
15. E. Hamburger, B. Cohen and R. Price, Preprint (1960).
16. N. Lassen and V. Sidorov, Nucl. Physics 19, 579 (1960).
17. D. Hughes and R. Schwartz, Neutron Cross Sections, Report BNL-235, Second Edition, USA, EC (1958).
18. V. M. Pankratov, N. A. Vlasov and B. V. Rybakov, Atomnaya Energiya 9, 5, 399 (1960).
19. N. A. Vlasov and A. A. Ogloblin, Nuclear Reactions at Low and Medium Energies [in Russian] (lzd. AN SSSR,
Moscow, 1958).
20. N. A. Vlasov et al., Zhur. Eksperim. i Teoret. Fiz. 37, 1187 (1959).
21. N. A. Vlasov et al., Zhur. Eksperim. i Teoret. Fiz. 39, 1618 (1960).
22. H. Tyren et al., Nucl. Physics 7, 1 (1958); H. Tyren and P. Isacsson, Canad. Conference (1960) p. 429.
23. N. A. Vlasov et al., Zhur. Eksperim. i Teoret. Fiz. 39, 1468 (1960).
All abbreviations of periodicals in the above bibliography are letter-by-letter transliter-
ations of the abbreviations as given in the original Russian journal. Some Or all of this peri-
odical literature may well be available in English translation. A complete Hist of the cover- to.
cover English translations appears at the back of this issue.
999
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
A SURVEY OF NUCLEAR-REACTOR DESIGN METHODS
G. I. Marchuk
Translated from Atomnaya Energiya, Vol. 11, No. 4, pp. 356-369,
October, 1961
Original article submitted June 22, 1961
INTRODUCTION
The wide application of high-speed computers in calculation has brought about conditions that are favorable
for the development of many regions of science and technology.
It has become possible to solve new and very complex mathematical and logical problems that were previously
considered to be unsolvable in practice because of the very great amount of numerical work involved in their solution.
In this connection, the questions concerned withthe mathematical formulation of a problem and the derivation of an
effective numerical logarithm for its solution are of particular importance.
The latest developments in numerical analysis have necessitated the review of logarithms for the solution of
many problems of mathematical physics, and the derivation of new logarithms based on the application of numerical
methods that are the best for the new computers available.
Analytical methods for the solution of problems have in many cases been replaced by more effective numerical
methods.
The achievements in the realm of numerical analysis have had a great influence on the development of the
theory and methods of calculation for nuclear reactors. After the book of S. Glesston and M. Edlund [1], in which
the elementary foundations of the theory of thermal-neutron reactors were described, there appeared the monograph
of A. D. Galanin [2], which was also devoted to questions on the theory of thermal-neutron reactors. The theoretical
foundations of nuclear energy have been most thoroughly described in the books by A. Weinberg and E. Wigner [3],
and also in the book by B. Davison [4]. The author of the present article has studied methods of performing cal-
culations for nuclear reactors in the monographs [5, 6]. A book has also recently appeared by R. Meghreblian and
D. Holmes [7], in which questions on the theory and method of calculation for nuclear reactors are considered.
Further developments in the theory and design of nuclear reactors were also published in the reports on the
First and Second International Geneva Conferences on world-wide investigations of atomic energy.
There is still theoretical value in the works of the earlier period of development of nuclear energy. These
results were obtained by E. Wigner [8], R. Peierls [9], R. Marchak [10, 11], N. N. Bogolyubov [12], I. I. Gurevich
and I. Ya. Pomeranchuk [13], H. Hurwitz [14, 15], A. Weinberg [16], and others.
A great proportion of the development of the theory of methods of calculation for nuclear reactors was carried
out in the Soviet Union.
The main progress.has occurred in the theory and design of thermal reactors, which is described by I. V.
Kurchatov [17], A. P. Aleksandrov [18], A. I. Alikhanov [19], D. I. Blokhintsev [20, 21], V. S. Fursov [22], I. I.
Gurevich and I. Ya. Pomeranchuka [13], S. M. Feinberg [23-25], A. D. Galanin [2, 26], V. V. Orlov [27, 28], and
others.
On the basis of these works, the scientific design was carried out for nuclear reactor projects for power stations
such as the First Atomic Electrical Generating Station, for the icebreaker 'Lenin', and for other purposes, and a
large number of experimental reactors were also planned for using various moderators and methods of heat-exchange.
The theory of and calculations for reactors for intermediate neutrons have been developed in the works of
A. I. Leipunskii, A. S. Romanovich, L. N. usachev [29, 30], V. Ya. Pupko [31, 32]. V. A. Kuznetsov, B. F. Gromov,
G. I. Toshinskii, V. V. Chekunov, V. V. Orlov [33], and others.
1000
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
A #
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
The theory of and calculations for reactors for fast neutrons have been developed and improved in the works
of A. I. Leipunskii [34-36], D. I. Blokhintsev [36-37], L. N. Usachev [29, 30], 0. D. Kazachkovskii [38], I. I.
Bondarenko [39, 40], S. B. Shikhov [41, 42], Yu. Ya. Stavisskii [34, 35], V. S. Vladimirov [43, 44], Yu. A. Romanov
[45], and others.
The works we have referred to formed the scientific foundation for the nuclear energy stations BR-2 and BR-5,
the construction of which has now been completed, and which came into active operation in 1956 and 1958 re-
spectively.
A review of the results of foreign scientific workers in reactor theory and design was given in the reports on
the First and Second Geneva Conferences.
The investigation of finite-difference equations approximating the integro-differential equations for,a reactor
is described in the works of A. N. Tildlonov and A. A. Samarskii [46, 47]. In these works there is a very full in-
vestigation of the problems of three-point finite-difference schemes, a practical logarithm is given for the most
efficient approximation to differential equations by finite differences, and the general properties of the solutions of
difference equations are established. The authors introduce the idea of integral accuracy for the solution of a finite-
difference equation relative to the solution of the differential equation, and prove theorems on the convergence of
the solutions of the finite-difference equations. It is shown, in particular, that difference equations obtained in the
class of discontinuous coefficients are of second integral order accuracy.
For the solution of three-point finite-difference equations of elliptic type, I. M. Gel? fand and 0. V. Lokutsievskii
(see (5)), A. S. Kronrod (see [2]), and also Stark [48] independently derived the method of linear factorization, which
can be used to reduce a boundary-value problem to the successive solution of three finite-difference equations of the
first order.
There is a method, interesting from the practical point of view, of solving the finite-difference equations in
a two-group approximation, proposed by G. M. Aderson-Verskii (see [2]). This method is based on the solution of
the boundary-value problem by using a solution of the Cauchy problem. The effective application of the method
is possible, due to the stability developed in the numerical scheme.
An essential advance was obtained in methods of solving the problems involved in the physical design of
reactors by using statistical methods, usually called Monte-Carlo methods. These latter methods, in their turn,
became feasible with the advent of computers with an even higher speed of operation.
The essence of Monte-Carlo methods, in their application to problems of nuclear physics, is the statistical
modelling of physical processes described by an integro- differential transfer equation. With this object in view,
certain random processes are considered, based on the elementary laws of behavior of particles, and the whole history
of a particle is traced, from the instant of its generation in the system being considered to its capture by or escape
from the system. The multiple repetition of this process, taking into account all possible probability situations,
makes it possible to accumulate the statistics necessary for obtaining an approximate solution of the problem.
The Monte-Carlo method for determining the critical reactor mass was developed by V. S. Vladimirov and
I. M. Sobol? [49].
A new approach to the solution of the above problem, based on the application of the adjoint equations for
the reactor, was developed by I. M. Gerfand, N. N. Chentsov and A. S. Frolov [50]. In these authors' work, an
algorithm was given for the solution of the problem by space-energy dependent operators.
A great amount of mathematical investigation in the theory of radiation transfer in general, and in the theory
of nuclear reactors in particular, was carried out by E. S. Kuznetsov [51-53] in collaboration with T. A. Germogenov
[54, 55], M. V. Maslennikov [56, 57], V. A. Chuyanov [58, 59]. L. V. Maiorov, M. G. Kriutikov, and others.
The problems involved in the physical design of reactors and methods of solving these problems have been
systematically investigated by the author of the present article in collaboration with his fellow-workers Sh. S.
Nikolaishvili [60, 61], V. V. Smelov [62, 63], E. I. Pogudalin [64, 65], V. P. Kochergin [65, 66], F. F. Mikhailus
[671, G. A. Ilyasov [65], and others.
The problems involved in nuclear-reactor calculations were also the object of investigations by A. S. Kronrod
(see [2]), N. Ya. Lyashchenko [25], V. K. Saul'ev [68], and others.
1001
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
The progress in numerical analysis has been widely discussed in scientific literature, and also at conferences
and meetings.
In the present survey, we will consider certain questions on the theory and design of reactors which have
generated wide interest in connection with the practical work of scientists in the Soviet Union.
The Main and Conjugate Equations
The most general mathematical formulation of the design problem for the critical regime of reactor operation
Is obtained from the integro- differential Boltzmann equation with the boundary condition corresponding to the outer
surface S of the reactor:
IMP + ET ? dv' Arc') (r, v') w(?0, v' ?> v) = 0; (1)
(r, Q, v) = 0 on S for n vs) = 0, (10)
(I)* (r, 12, v) = 0 on S for Uri >0. (.3.)
The simplest conjugate reactor equations were introduced by E. Wigner in [3]. For the single-velocity problem,
the conjugate equation was formulated in a very general form by K. Fuchs [69] and N. A. Dmitriev (see [30]). The
most complete conjugate reactor equation, taking moderation into account, was obtained by L. N. Usachev [30].
The latter also gave a physically illustrative interpretation of the conjugate function as the 'value* of the neutrons.
The theory of conjugate equations was further generalized by B. B. Kadomtsev [70] for the nonhomogeneous equations
of particle transfer. In the work by the author and V. V. Orlov [27], a method was developed for obtaining conjugate
equations for a wide class of nonhomogeneous linear problems, and the theory of disturbances for various linear
functionals was also formulated in general form.
In the cases of the calculation of critical mass, the space-energy distribution of current, and the neutron value,
the exact solution of the kinetic Eq. (1) for the boundary condition (2) is very difficult. In the majority of cases
therefore, only approximate solutions are sought. Among the approximate methods that are useful in this work, an
1003
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
especially important example is the diffusion approximation. The essence of the method lies in the fact that the
solution of the problem (1) and (2) can be written as a series of spherical harmonics, and the first two terms of the
series can be used as an approximation (101 (the Pr-approximation).
Therefore,
(r, v) [To (r, v) + 3S/ch. (r, 0], (12)
where the function vo is the total neutron flux through the surface of a unit sphere with center at the point r, and
fits the vector flux of neutrons at the point r.
We expand the function w (?0, v' v) in a series of Legendre polynomials and retain only the first two terms.
We then have
v' ?> v) = [wo (v' v) 31towl (v'
(13)
If we substitute the expressions in (12) and (13) in Eq. (1), we obtain the system of integro- differential equations
V(pi+ Zwo?
1
?Eq),. ?
3 o
dv'wo (v'
dv'tvi
v)
---> v
X w(r, v') = 0;
X (pi (r, v') = O.
(14)
The boundary condition for the system of Eqs. (14) is obtained, according to R. Marshak [101, in the form
24:p1n To.= 0 on S. (15)
The system of Eqs. (14), together with the boundary condition (15), form the closed system of basic equations for a
reactor for the diffusion approximation.
The system of adjoint reactor equations, for the diffusion approximation, can be obtained similarly, and is
_vc__Eztp:_sdv,w0(v.._>v,) X IT: (r, tf) = 0; 1
dv'tvi X (v --> v') (r, V') = 0
The boundary condition for the system of Eqs. (16) is
(16)
2irg`n cp: = 0 on S. (17)
The set of Eqs. (16), together with the boundary condition (17), form the closed system of adjoint reactor equations
for the diffusion approximation.
Similar equations and boundary conditions that can be obtained for various more accurate approximations than
the above can also be obtained by applying the spherical-harmonic method.
In connection with the fact that it is very difficult to solve the main and adjoint reactor equations in their
most general form, and that the solution can be obtained only in very rare cases, the necessity arises of developing
1004
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
various approximate methods. One of the most widely-used approximate methods is the multigroup method. This
method is based on breaking up the whole neutron-velocity range in the reactor into partial intervals, in each of
which it is assumed that the physical parameters are constant. Then, in each of the groups, in which the physical
parameters are assumed to be constant, we can integrate the Eq. (1) within the limits of the given energy group,
and use as unknowns the integral neutron flux for the groups. We formally arrive at a multigroup system of equations
for the reactor. Thus, for example, the multigroup system of basic equations in the 1-approximation is [5, 6]
with the condition that
i ?
V(Pi Zo% ? E zowt, =0;
I
3
-- V o
1 . j ? x---1
W] L ZIT
2win ? (pio 0 on S.
The multigroup system of adjoint equations becomes
-vq)1)+150C 10(PV = 0;
-1?Tti'+liwi'- I ;Tv= ?'
3
where the boundary condition at the outer surface is
2ctin (pti = 0 on S.
(18)
(19)
(20)
(21)
5
It is necessary to use perturbation theory to obtain the system of multigroup constants Zo, and functions
1-4.J t-4.J
For this, we require that in the transition from the problem (1) and (2), which we will call the un-
perturbed problem, to the multigroup perturbed problem (16) and (17), the critical dimensions of the reactor or any
other characteristic parameters of the problem remain constant.
We thus arrive at the following formulas for the group constants [5, 61:
0)
S drce04 5 dolco
.
G1 o j-i
v ?
rcpt)i 5 dvco
Gn
Cl,
5 drei) 5 dOnpi
n
"1
drotpl) .S dmpi
Gn v"
I j
Zo=
Avi 5 drq41 5 cluTo 5 dv'wo(v v')
an vi_,
01
drcp:ji 5 clop?
(22)
1005
I Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
/1
Sdrq Amp,. dv'vvi(v v')
Gn v1-1 v1-1
S
dry' dvpi
Gn
(22)
In the formulas (22), it has been assumed that we know the neutron-flux spectrum at every point of the reactor
and the multigroup neutron value. Strictly speaking, however, neither the neutron flux or the multigroup neutron
?
value is known beforehand, so that the exact values of the group constants, defined by the formulas (22), are unknown.
The above formulas can nevertheless be successfully applied in the approximate calculation of the group
constants. Thus, for example, in the active zone of the reactor the mean of the constants can be obtained by taking
into account the spectrum of the equivalent reactor without neutron reflectors, and in the reflector the mean can
be obtained from the integral neutron flux obtained beforehand from the balance relations for the reactor. Such an
approximate method has been developed by S. B. Shikhov [41, 42].
There are other methods of averaging that take into account more precisely the concrete features of a reactor.
After the approximate neutron spectrum has been found, the algorithm for averaging the physical, constants of a
reactor with one-dimensional geometry is as follows. In the formulas in (22), we set 9; =1 and cp =k, where
k is a vector in the direction of the normal to the coordinate surface, and we obtain a simple system of group con-
stants which can be used in the solution of the adjoint multigroup problem (2c and (21). After this, we again cal-
culate the group constants, etc. This process converges very rapidly, so that after two or three iterations the group
constants are sufficiently accurate. After this, the set of group constants that have been obtained is used in the
solution of the multigroup system of reactor Eqs. (18) and (19).
If the group interval is sufficiently small for the neutron flux to vary only slightly within the limits of each
group, then the averaging of the physical parameters is not essential, and in this case we can determine a universal
system. of group constants, independent of the neutron spectrum in the reactor. If, on the other hand, the group
intervals are large, then for intermediate and fast reactors it is necessary to use the averaging method described
above.
We note, in conclusion, that in spherical-harmonic methods of higher accuracy than the Pr approxithation,
we can obtain group-average formulas similar to those given in (22). To obtain the mean constants from these
formulas, it is necessary to know the neutron spectrum at every point of the reactor. The spectrum is obtained by
using various approximate methods, among which is the Pr approximation method. ,
Numerical Spherical-Harmonic Methods of Solution
Among approximate methods of solving the kinetic equations, the spherical-harmonic method occupies a
conspicuous place. This method was developed by G. Vik, and improved by R. Marshak [10], L. N. Usachev [291.
?B. Devison [4], G. Mark [71], V. S. Vladimir [43, 44], and others. The basis of the method is to obtain the solution
of the kinetic equation in the form of a Fourier series of spherical harmonics. To obtain the unknown Fourier co-
efficients, we have an infinite system of differential equations (with derivatives with respect to the geometrical
coordinates).
We consider, for simplicity, a one-velocity kinetic equation, which is an element of the group calculations.
If it is assumed that the neutron dispersion in the laboratory system of coordinates is obtained as a function g (14),
then the neutron-transport equation for the plane-parallel problem will have the form
Is ,
tt + IT = -2- S g T (z, EL') +1(z, P.),
1006
(23)
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
where 0,0 = pp, -EV ? p,2 ? 2 cos a with a as the azimuth.
We will seek a solution of the Eq. (23) in the form
it) E (2/-1-1) cpt (z) (?),
i=o
where P1 (?) is a Legendre polynomial. As a result, we obtain the system of ordinary differential equations
where
m d(Pm-1 + 1 d(Pm..i. (2m + I) Emopm = (2m +!1) nt'
dz dz
En,. ?gmEs;
? and gm and fm are Fourier coefficients in the expansion of the functions g (?) and f (z, p) in the series of
Legendre polynomials.
On the outer boundary of the region, we use Marshak's conditions
(24)
(25)
dittp,2i+1 = 0 on S (i = 0, 1,2, ...). (26)
?t
When we substitute the solution (24) in the condition (26), we arrive at the boundary conditions for the Fourier co-
efficients
E 0 on S,
(27)
where aim are given numbers.
For the solution of the system of ordinary differential Eqs. (25) with the condition (27), we use the apparatus
of matrix calculus. With this in view, we introduce the vectors
(PI
(I) =
11)2
=
(1)3
f0
f
f2
, F =
f3
? ? ?
and consider separately the Eqs. (25) for even values of i'm"? 0, 2, 4, and for odd values lj, 3, 5, . . .
We then obtain the matrix equations
thlr
--aq) f ;
dz
'
p bJ =F,
(28)
1007
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
where a, 8 a and b are known matrices, determined from the coefficients of the Eqs. (25). At the same time,
the boundary condition can be written
+ Bj = on 8.
(29)
The final problem consists of solving the system of Eqs. (28) with the boundary conditions (29).
By the application of methods that are well established in the calculus of finite differences, the Eqs. (28), in
the case of piece-wise continuous coefficients, can be written as the single three-point, matrix, finite-difference
system of equations
erp,,+ CAD = ?
(30)
where k is the number of the point on the z-axis, Bk and Ck are matrices, and gk is a vector.
The Eq. (30) is resolved by the matrix-factorization method into the three difference equations of the first
order:
It
= (P.,
z.,+1= (z, 4- g.,);
(Ph = ci7-11-1 (13k.iwk+1 + zh+i)?
(31)
The set of Eqs. (31) is supplemented by given 'initial" conditions obtained from the boundary conditions [5, 6].
The matrix-factorization method is the most effective numerical method of solution of spherical-harmonic
equations, and it can be used to solve any problems in one-dimensional plane, cylindrical, or spherical geometries.
As an illustration, we note that the kinetic equation for an infinite cylinder in the P3- approximation, for five zones
and 60 points, can be solved by an electronic computer in 7 sec. The critical dimension of a spherical reactor, in
the multigroup P3-approximation, is obtained by a computer in only four times the calculating time needed for the
P1- approximation.
It can evidently be asserted, at least for one-dimensional problems, that the solution of kinetic equations by
the spherical-harmonic method, in combination with numerical, matrix-factorization method, yields a very effective
algorithm.
We should also mention another numerical method of solving the spherical-harmonic equations, developed by
S. K. Godunov, and presented at the All-Union Conference on approximate methods in Moscow in 1960.
This method consists of the solution of the boundary-value problem for a system of spherical-harmonic
equations, by using problems with initial values. A stable numerical algorithm is obtained by applying a special
algorithm for the systematic smoothing of rounding errors arising in the calculation.
Small-Group Approximations
In the application of electronic computers in the design of nuclear reactors, certain difficulties are often en-
countered, connected with the limitations of the machines available. First of all, this applies to the memory of
the machines and their speed of operation. In this connection, the question arises of the most convenient mathematical
arrangement of a problem in given circumstances, and the arrangement that will ensure that the solution will contain
the maximum possible amount of information. We note that the information must be not only complete, but also
reliable.
In the physical calculations for a reactor, the multigroup operation is desirable for two- or three-dimensional
cylindrical reactors, carried out with the fewest possible steps, and taking into account the various physical effects
in the Pa-approximations. It is clear, however, that the contemporary numerical computers cannot carry out the
solution of the problem we are considering completely, even with the progam needed for the design. The solution
of these problems demands such a great amount of machine time that the calculation is neither economically
rational nor justified.
1008
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
The physical calculations for a reactor can be broken up into several very simple problems, the solution of
which is not difficult. The set of solutions of the corresponding one-dimensional problems, in conjunction with
perturbation theory, can be used to yield very complete information on the details necessary in the design of a
reactor. The complex problem is thus replaced by a collection of more or less simple problems, for which the
solutions can be obtained by using electronic computers. This means that in each concrete case we must deter-
mine the choice of simple problems that will provide the information necessary for the design of the projected
system.
In connection with what has been described, a verylive question at the present is that of the small-group
system of reactor equations.
In what follows, we will consider the one-group and the three-group methods.
The critical mass of a reactor can be found by using the effective one-group theory and the formulas (18) to
(31) with the assumption that all the neutrons are combined into one group. For this, we can obtain a system of
effective one-group constants by solving the multigroup problem in the Pr approximation. The resulting one-group
kinetic equation will be used in the following for making more precise the calculation of critical parameters of
a reactor in the Ps- approximation. The simplest method of taking the kinetic effects into account is the
following. We solve the multigroup system of equations of the reactor in the Pc-approximation, then obtain the
mean constants for the one-group theory, and finally solve the one-group problem in the P3-approximation. A
comparison of the calculations of the critical masses in the Ps- approximation using the multigroup method with
the corresponding calculations for the one-group theory indicates a very high accuracy for the one-group calcula-
tions, which is completely satisfactory for all practical purposes.
If the one-group theory can be used only in making the value of the critical reactor mass more precise, then
the three-group method will give an essential improvement in the accuracy of the energy-output field in the active
zone of the reactor. In the method we are considering, the fast neutrons are combined into one group, the inter -
mediate-energy neutrons into another group, and the thermal-energy neutrons into a third. Thus, the solution of
the problem again begins with the solution of one-dimensional, multigroup, reactor equations in the Pr approxi-
mation. The neutron flux thus obtained is subsequently used in the calculation of the effective three-group con-
stants. By using the three-group system of equations, we can improve the accuracy of calculation of the critical
mass and the energy output in the P3-approximation and also carry out the physical calculations for the reactor
in two-dimensional or three-dimensional geometry.
The one-group and the three-group methods of reactor calculation, using effective methods for averaging
the constants, was proposed by the author of the present article [5, 6], and subsequently developed by him in
collaboration with E. I. Pogudalin and V. P. Kochergin.
Heterogeneous Reactor Design
The theory of heterogeneous-reactor design, in the case of thermal neutrons was begun by S. M. Feinberg
[23], A. D. Galanin [2], [26], G. A. Bat [72], and others. This theory can be used to carry out the direct calculation
of the critical parameters of a heterogeneous reactor and the neutron spectra. The theory is based on the following
assumptions [23].
The field of the thermal neutrons in the region of a core containing the active isotope is axially symmetric,
so that the neutron sources and sinks can be considered to be linear. To describe the diffusion of the thermal
neutrons between the cores, we can apply the elementary diffusion equation. Finally, the absorptive capacity of
the uranium cores is characterized by the logarithmic derivative of the neutron flux at the core surface. When
these assumptions are used, the problem of critical-mass calculation for a heterogeneous reactor, using influence
functions, reduces to a system of homogeneous, linear, algebraic equations.
In the calculations of the critical mass of a reactor ?we can also use effective homogenization methods, in-
cluding the calculation of the effective constants for fast, intermediate, and thermal neutrons.
We consider, first of all, the calculation of effective constants for the energies of the fission spectrum. In
this case, it is of particular importance to calculate the capture-fission cross section for U238, since this cross section
can be used to calculate the multiplication coefficient of the reactor for fast neutrons. The calculation of the
effective capture-fission neutron cross section is carried out, taking into account the mutual shielding of the uranium
1009
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
cores, by using the method of successive collisions [1]. Other constants are calculated using various homogenization
methods [62, 73, 741.
Resonance effects in the intermediate-energy range are taken into consideration by using effective resonance
integrals, and the cross sections, varying smoothly with the energy, are subjected to the process of formal homo-
genization. If the core-effect exists, i.e., if the neutron flux varies considerably between the reactor cells, then we
must use effective homogenization methods.
In the calculation of the one-group constants for the thermal group of neutrons, we first of all consider the
problem of finding the space-energy neutron distribution in a reactor cell, taking the thermalization into account.
The calculation is carried out in the P1-approximations. The one-group averaging of the constants is then performed.
The resulting constants are used for the one-group calculation of the integral spectrum of the slow neutrons in the
P1 and P3-approximations for the volume of a cell. The cross sections relative to a cell are finally averaged, to
obtain the effective constants for the equivalent homogenized reactor. If the calculations in the P1 and P3- approxi-
mations, in the one-group model, lead to an essential difference in the calculations for the homogenized constants,
then it is necessary to use the calculation of the space-energy distribution of the neutrons in the P3- approximation,
taking into account the thermalization.
The numerical methods of calculating the spectra of slow neutrons were developed by the author and V. V.
Smelov (75, 621. Further theoretical results on these questions can be found in the works of V. V. Smelov, L. V.
Maiorov, and others.
In the results of A. D. Galanin [2, 761, some theoretical developments are given concerning the spectrum of
slow neutrons in a heterogeneous medium, and important estimates are given, which agree well with the results of
numerical calculations.
After the effective homogenized constants have been obtained, the calculations for a heterogeneous reactor
are the same as those for a homogeneous reactor, and can be carried out with the program used for the latter.
The Calculation of the Compensating Capacity of the Controlling Rods
The calculation for the compensating system of a reactor is the most complex problem of physical design.
This problem can be solved, more or less simply, only in the case when the compensating rod is completely in-
serted into the reactor and there are no end reflections (the latter can be replaced by effective additions to the
height), and the rod coincides with the axis of the reactor. In this case, the problem reduces to a one-dimensional
problem, and can be solved in the multigroup approximation by using effective boundary conditions at the surface
of the compensating rods. Let the introduction of a rod change the characteristic number X =1/Keff of the problem
by the quantity 6 x0. In the diffusion approximation for a cylindrical core, the formula for 6 X.0 was obtained by
L. N. Usachev:
21tr0 Ni Di ?irAi
= S drQQ* wi (Po To
i
where y is the logarithmic derivative on the interval of the group [77](1/ ? ? v.). ro is the radius of the core, G is
j
the reactor volume, Q (r) is the total number of secondary neutrons and Q?(r) is the neutron fission value. The
quantity 6 X for a rod located at a distance rj from the center can be estimated by using a formula from perturba-
tion theory obtained by L. N. Usachev [29]:
Q* (rJ)Q (ri)
"i ?0 Q* (0)Q (0) '
where Q. (r) and Q(r) are unperturbed functions, obtained in the absence of the rod. We note the point that per-
turbation theory yields a value for the quantity (5 X j only, and cannot be used to calculate the neutron flux or the
function Q(r) in the presence of a rod. The calculation of the neutron flux, even in the very simple case where the
rod is located at a distance r from the center of the cylinder, involves the solution of a two-dimensional problem
in the (r, 0) plane.
1010
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
We arrive in a similar way at the necessity of solving a two-dimensional problem when a rod, situated at
the point r = 0, is partly withdrawn from the reactor. The same case occurs when we consider a compensating
cylinder, or a set of compensating cylinders situated symmetrically relative to the origin.
The calculation is still more complicated when a system of compensating rods in a reactor is, considered for
various methods of withdrawing the rods from the reactor with the consumption of the active part of the uranium.
In this case, it is necessary to carry out calculations for a series of three-dimensional reactors with various rod
positions.
The above two-dimensional and three-dimensional problems can be solved by an effective three-dimensional
method in the Pr approximation with effective boundary conditions at the surface of the compensating rods or
cylinders.
The Problem of the Physical Design of a Reactor
The advent of high-speed computer techniques in numerical analysis has provided the necessary prerequisite for
the physical calculations for nuclear reactors with any spectra, and with various complicated setups for the active
elements and the moderation. Here, it is necessary to formulate a single method which can be used to carry out
the calculations for a homogeneous reactor with any possible spectra from fast to thermal. Such an approach is
naturally only possible when the method developed takes into account all the basic features of the moderating
mechanism for various energies and also takes into account the cross sectional structure of the nuclear processes.
In the majority of cases, the calculation of the critical mass and the neutron spectrum is possible in the PI-
approximation. An exception is the type of reactor in which the mean free neutron Path is comparable to the
characteristic dimensions of the reactor. The calculations in the case of this type of reactor in the P1-approximation
can lead to gross errors in the determination of critical mass and of neutron spectrum. In this case it is necessary,
In the calculations, to use a more accurate approximation than the P1- approximation. For such calculations we
must use, for example, the Ps- approximation.
The performance of the serial multigroup calculations for a reactor in the P3-approximation on a computer can
use considerable amounts of machine time. The calculation of critical masses can therefore be performed by using
effective one-group theory, programmed for the calculations in the Pi-approximation.
It is very important in the calculations to take account of the resonance structure of the cross sections, and
also the thermalization of the neutrons.
All the above particulars of the reactor must be taken into consideration in obtaining the mathematical
algorithm for solving the problem of calculating the critical dimension of the reactor. In what follows, we will
formulate some algorithms for the solution of the basic problems involved in the physical design of nuclear reactors
with various spectra. The aim of these algorithms is to lead to programs for electronic computers.
In the USA, programs for multigroup calculation are based on algorithms developed by G. Hurwitz and R.
Erlich [48, 78].
In the calculation of the critical dimension of a reactor in the PrapproximatiOn, the neutrons of all possible
energies are divided up into groups. Neutrons with energies less than vgr are combined into one group, to which
effective constants are ascribed. Both elastic and nonelastic effects should be taken into consideration in the neutron-
dispersion process. In the neutron moderator, we must consider the resonant character of the capture and scattering
cross section. The neutrons arising during the fission process are distributed in groups corresponding to the fission
spectrum. The zero-order and first-order moments must be taken into account in the neutron-scattering function.
The system of basic and adjoint equations for the reactor is solved by using the method of successive
approximations [5].
In the one-dimensional and two-dimensional cases, the solution of the reactor equations is carried out for
each group by using finite differences. In the calculation, it is convenient to use the simplest possible difference
schemes, in which the discontinuities in the coefficients in the equations coincide with the points used in the cal-
culations [5, 6, 46, 47]. The finite-difference equations for one-dimensional regions are solved by the factorization
method, while for two-dimensional regions various relaxation methods are applied [68, 79]. A very general algorithm
for the calculations arising from two-dimensional reactors was obtained by T. A. Germogenov.
1011
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
Declassified in Part - Sanitized Copy Approved for Release 2013/09/25: CIA-RDP10-02196R000600070002-0
In those cases where the calculation of the critical mass of a one-dimensional reactor in the Pr approximation
is not satisfactory, we can use the P3-approximation. Multigroup systems are formulated for the basic adjoint
equations for the given one-dimensional geometry in this case.
In the majority of cases, the critical mass can be satisfactorily determined by using the first two moments
in the scattering function. We must mention, however, that the calculation of the neutron spectrum at large distances
from the active zone should preferably include four moments, since in this case the anisotropy of the neutron flux
must be taken into account more accurately.
In calculations for reactors with dimensions that are not very large, it is necessary to solve the multigroup
kinetic equations. In the transport approximation, this problem reduces to the successive solution of single-velocity
Boltzmann kinetic equations. For spherically symmetric regions, we have the equation
and the boundary condition
Here
am 1 u 2 am
?alL; +p = Q (r)
(32)
(r, ti) = 0 for It < 0. (33)
Q (r) = 1(r);
(34)
where f (r) is a given function.
For the solution of the Eqs. (32) to (34), we can use Vladimirov's method of characteristics [43, 44], or the
Sn- methodof Carlson [80]. Both these methods are based on iteration processes, in which each iteration involves
the calculation of the quantity Q(r) from its value obtained in the previous calculation.
The method of characteristics uses the change of variables
x = rp.; y = r -171 ?
where the partial differential equation (32) is reduced to an ordinary differential equation of the first order, which is
solved for y = const. In the solution of the above equation, Vladimirov applies a very original numerical scheme,
which permits a very efficient use of the results of the calculation. After obtaining the solution cp (r, ?) at the
points of a special grid, the function Q(r) is calculated. The process continues in the same way.
The essence of the Sn-method of Carlson is the numerical solution of the Eqs. (32) - (34) using a finite-
difference method, in which the interval ?1
co ra 4, Cr) 404
t
0/ ? ctl C\1 t-
(1)
(1)
-%
cd
8
C.)
?.-1
E
0
0-.