SOVIET ATOMIC ENERGY VOLUME 20, NUMBER 5

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP10-02196R000700030001-4
• Release Decision: RIFPUB
• Original Classification: K
• Document Page Count: 115
• Document Creation Date: December 23, 2016
• Document Release Date: April 1, 2013
• Sequence Number: 1
• Publication Date: May 1, 1966
• Content Type: REPORT

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Volume 20, Number h
SOVIET
ATOMIC
ENERGY
ATOMHAfi 31-1E13111H ?
(ATOMNAYA iNERGIYA)
TRANSLATED FROM RUSSIAN
CONSULTANTS BUREAU
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SOVIET
ATOMIC
ENERGY
Soviet Atomic Energy is a cover-to-cover translation of Atomnaya
Energiya, a publication of the Academy of Sciences of the USSR.
An arrangement with Mezhdunarodnaya Kniga, the Soviet book
export agency, makes available both advance copies of the Rus-
sian journal and original glossy photographs and artwork. This
serves to decrease the necessary time /lag between publication
of the original and publication of the translation and helps to im-
prove the quality of the latter. The translation began with the first
issue of the Russian journal.,,
gditorial Board of Atomnaya Energiya:
Editor:, M. D. yilliqnshchikov
DeputY Director, Institute of Atomic Energy
imeni I. V. Kurchatov
Academy of Sciences of.the USSR
Mosaow, USSR
Associate Editors: N. A. Kolokol'tsov
? 7 N. A. Vlasov
A. I. Alikhanov
A. A. Bochvar
N. A. Dollezhal'
V. S. Fursov
I. N. Golovin
V. F. Kalinin
A. K. Krasin .
A. I. Leipunskii
V. V. Matveev
M. G. Meshcheryakov
P. N. Palei
V. B. Sherchenko
D. L. Simonenko
V. I. Smirnov
A. P. Vinogradov
A. P. ZefiroV
Copyright? 1967 Consultants Bureau, k division of PlenuAl Publishing Corpora-
tion, 227 West 17th Street, New York, K. Y. 10011. All rights reserved. No article
contained herein may be reproduced for any purpose whatsoever without per-
mission of the publishers.
Subscription
(12 Issues): $95
Order from;
CONSULTANTS BUREAU
Single Issue: $30
Single Article: $15
227 West 17th Street, New York, New York 10011
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SOVIET ATOMIC ENERGY
A translation of Atomnaya energiya
Volume 20, Number 5 May, 1966
CONTENTS
Engl./Russ.
"X" (1956-1966)
423
378
Economic Incentives in the Nuclear Power Field?V. V. Batov and Yu. I. Koryakin
424
379
Heavy-Current Accelerator Based on a Transformer?E. A. Abramyan and V. A. Gaponov
431
385
Experimental Determination of the Radiation Quality Factor Near High-Energy Accelerators
?V. N. Lebedev, M. Zerchinskii, and M. I. Salatskaya
439
392
Helical Magnetic Configurations with Minimum B?N. M. Zueva and L. S. Solov'ev
444
396
Paramagnetic Effect under the Influence of High-Frequency Pressure and Electron Paramagnetic
Resonance in Plasma?V. M. Glagolev, I. N. Khromkov, and N. S. Cheverev
452
401
Optical Excitation and Ionization of Fast Hydrogen Atoms?D. P. Grechukhin, E. I. Karpushkina,
and Yu. I. Sokolov
459
407
The Effect of Certain Cycle Parameters on the Efficiency of a Nuclear Gas Turbine Unit
?E. F. Ratnikov and M. V. Shustov
464
412
Distribution of Fast Fission Neutrons along Straight, Cylindrical Ducts in Water
?V. P. Mashkovich, A. N. Nikolaev, V. K. Sakharov, B. I. Sinitsyn, and S. G. Tsypin. . . .
469
416
Theory of Azeotropic Rectification with Steam, Exemplified by the System Tributyl Phosphate
?Carbon Tetrachloride?B. Ya. Zil'berman, V. N. Komarov, and M. F. Pushlenkov
473
419
ABSTRACTS
Analysis and Generalization of the Correlation Method for Measuring Particle Lifetime
Distributions in a Physical System?V. G. Zolotukhin, A. A. Kutuzov, D. L. Broder,
L. P. Kham'yanov, B. A. Efimenko, and A. S. Zhilkin
477
422
Calculation of Yield and of Mean-Square Angle of Deviation for Positrons in the Penetration
of Thick Foils by Electrons?A. V. Bautin, and 0. S. Koifman
479
423
Dependence of Buildup Factor on Detector Position Outside Shielding?Yu. A. Kazanskii,
V. I. Kukhtevich, V. I. Popov, V. V. Tarasov, and B. P. Shemetenko
481
424
A Method for Computing Heat Transfer Coefficients for Longitudinal Flow of Liquid Metals
through Fuel Element Bundles?M. Kh. Ibragimov, and A. V. Zhukov
y -Ray Shielding of Artificial Stone?V. B. Dubrovskii, A. K. Shreiber, A. F. Mirenkov,
and V. N. Solov'ev
y -Ray Penetration through Joints of Builtup Concrete Shields?V. B. Dubrovskii,
Yu. S. Ryabukhin, A. F. Mirenkov, and V. N. Solov'ev
483
484
486
425
425
426
Equipment for Radiochemical Processes with a Reaction Vessel Giving a Uniform Temperature
Field?L. S. Polak, P. Ya. Glazunov, B. N. Parfanovich, G. G. Ryabchikova,
V. E. Glushnev, and V. T. Popov
488
427
LETTERS TO THE EDITOR
A Sector Cyclotron with Magnet Poles of Diameter 685 mm?A. G. Alekseev, V. N. Barkovskii,
Yu. G. Basargin, V. N. Vasil'ev, R. N. Litunovskii, 0. A. Minyaev, V. N. Nikolaev,
and A. V. Stepanov
490
429
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CONTENTS
Measurement of Fast Neutron Absorption Cross Sections with a Resonance Detector in Water
(continued)
Engl./Russ.
?Yu. Ya. Stavisskii et al
493
431
Analysis of Material Composition by Inelastic Scattering of Fast Charged Particles
?S. S. Vasil'ev, T.N.Mikhaileva, Yu. A. Vorob'ev and D. L. Chuprunov
496
432
Measurement of Large y -Ray Doses and Fluxes by Photoactivation of Isomeric Nuclear States
?I. A. Abrams, L. L. Pelekis, and I. Ya. Taure
500
434
Effect of y -Irradiation on Scale Formation?V. N. Vasina, V. N. Aleksandrova,
and V. V. Gerasimov
502
435
Apparatus for Oscillator Measurements in a Reactor?A. I. Efanov, L. V. Konstantinov,
V. V. Postnikov, I. P. Sadikov, and M. P. Sokolov
504
437
Cross Section Averaging in the Thermal Region for Media Containing Zirconium Hydride
?L. M. Gorbunov, F. M. Mitenkov, a B. Samoilov, and V. V. Farmakovskii
507
438
Thermionic Emission of Uranium Dodecaboride ?S. V. Ermakov and B. M. Tsarev
509
439
Effect of Ultrasound on the Plasticity of High-Boron Stainless Steel?L. E. Al'shevskii,
Yu. S. Kuz'michev, L. M. Kurochkina, I. S. Lupakov, V. E. Neimark, and I. I. Teulin . . .
511
440
Ionization-Mechanical Detector for Ionizing Radiations?O. A. Myazdrikov, V. N. Demidovich,
and A. P. Suslov
514
442
Express Method of Determining the Concentration of an RaA Aerosol and the Latent Energy
in the Air?N. P. Kartashov
517
444
How to Calculate Changes in the Concentration of a Radioactive Isotope in the Waters
of a Noncirculating Reservoir with Isotope Absorption by the Bottom Layer
V. M. Prokhorov
522
448
NEWS OF SCIENCE AND TECHNOLOGY
Conference on Research Reactor Physics and Technology?G. Zhemchuzhnikov
525
450
[IAEA Symposium: Use of Radioisotope Techniques in Industry and Geophysics (Warsaw,
October, 1965)?F. A. Alekseev, D. A. Kozhevnikov, R. A. Rezvanov, and A. I. Kholin. . .
451]
[IAEA Conference: Problems of Radioactive Waste Fixation (Dubna, December, 1965)
?B. S. Kolychev
452]
Seminar at the USSR National Exhibition?T. I. Nezhel'skaya
527
454
The Unit of Measurement for Biological Dose of Ionizing Radiation?Yu. V. Sivintsev
528
455
NOTE
The Table of Contents lists all material that appeared in the original Russian journal. Items origi-
nally published in English or generally available in the West are not included in the translation and
are shown in brackets. Whenever possible, the English-language source containing the omitted items
is given.
The Russian press date (podpisano k pechati) of this issue was 5/14/1966.
Publication therefore did not occur prior to this date, but must be assumed
to have taken place reasonably soon thereafter.
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,M211,41.W2VR,M$:
***:
The periodical Atomnaya gnergiya is ten years old. It is only two years younger than the worldwide nuclear
power industry itself. Its pages have reflected the progress of nuclear power?from 5000 kilowatts at the world's first
power station to 10 million kilowatts, from laboratory experiments to completion of work on several types of scaled-
up nuclear power installations competitive economically with "classical" electric power generating stations. The
solution of the fundamental problem of fuel breeding -i including Th and U238 in nuclear fuel and expanding fuel re-
serves a thousand-fold, is embodied in the Soviet Union, in the construction of a fast reactor with industrial power
capability.
The third issue of our periodical (1956) published a report by Academician I. V. Kurchatov which set the basis
for the wide-open worldwide discussion of work on controlled thermonuclear fusion of light elements. The period-
ical has shed light on the subsequent development of this work and the discussion of results of these endeavors at
many international conferences. The complexity of the problem, obvious from the very outset, is now understood
in a more concrete and more fundamental way. Even though the most optimistic forecasts have not been validated
by the decade elapsed since then, confidence in ultimate succesz has broadened and gained strength.
The periodical's pages have also reflected achievements in the field of accelerator practi::::, rniclear chem-
istry, nuclear geology, the production and application of isotopes, radiation safety and health physics.
The volume of information published over the ensuing years since the start of our publication fills almost
1400 printing folios.
The editorial staff of the periodical takes this occasional to greet its contributing authors, readers, and all
those involved in nuclear power and related fields of science and industry, and to wish them renewed creative suc-
cesses accelerating technical progress.
Translated from Atomnaya gnergiya, Vol. 20, No. 5, p. 378, May, 1966.
423
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ECONOMIC INCENTIVES IN THE NUCLEAR POWER FIELD
V V. Batov and Yu. I. Koryakin UDC 338.4:621.039.516
Payments on fixed productive capital and on circulating capital in nuclear power installations are re-
considered in the light of decisions adopted at the September (1965) Plenum of the Central Committee
of the Communist Party of the Soviet Union. The discussion centers on circulating capital tied up in
nuclear fuel procurement and inventory. Qualitative considerations on rates of payment for productive
capital in nuclear power are outlined. Equations are derived for calculating the specific payment re-
lating to reactor performance data.
The basic decision adopted at the September (1965) Plenum of the Central Committee of the Communist
Party of the Soviet Union on payability of productive capital assigned to fuel-cycle enterprises prompts a review of
these economic incentives within the context of the nuclear power picture. The decision on the quantitative value
of profit write-off in budgeting, as related to the value of basic capital and circulating capital tied up in nuclear-
fueled power generating stations and in other fuel-cycle installations, is of enormous importance for the nuclear
power industry of the USSR, and especially for nuclear electric power stations producing marketable electric power
and characterized by certain specific costs features. These costs features are determined primarily by the familiar
economic dualism inherent in fuel inventorying which exhibits certain features of fixed capital on the one hand,
viz., in the way it functions (in view of the long operating period TkAp where Tk is the length of the reactor run,
co is the utilization factor of the power station installed capacity), and the cost (reaching 40% of the direct capital
investments in nuclear power stations), and on the other hand exhibits certain features of circulating capital, viz.
in the way costs are assigned to the salable commodity (electric power) over the production period and the uncom-
pleted nature of the production (the spent fuel serves as a semifinished product useful for further production opera-
tions, in view of the fissionable isotopes remaining and accumulated in the fuel). Nevertheless, assigment of fuel
inventory to circulating capital (i.e., long-term circulating capital) would still be a more correct procedure, espe-
cially if we bear in mind that, in contrast to the fixed capital tied up in the nuclear generating station, the fuel
inventory is essentially a less conservative item in its initial cost relationship as a part of the nuclear power station,
compared to other parts and components of the power station, and the fact that fuel inventory value is capable of a
wide range of variation during the operating process. ' In effect, nuclear power development practice has demon-
strated that there is not a single operating power reactor in which each subsequent fuel inventory did not differ
markedly from its predecessors in regard to engineering costs. Moreover, the fuel inventory is such a flexible and
dynamic part of the reactor picture that its costs characteristics vary markedly even during the fuel dwell time in
the reactor (during the reactor run). A highly intriguing and unique property of the fuel inventory, in the costs sense,
a property bearing little resemblance if any to other forms of production, is its ability in certain cases (as for in-
stance in fast breeders) to increase in value over the initial cost during a single reactor run. This can of course be
held accountable to the buildup of secondary fissionable material of high power-producing value. A consequence of
this property of the fuel inventory is, in some instances, the approximation of the fuel components of the nuclear
power generating costs to zero or even the appearance of a paradoxical negative fuel component in the nuclear elec-
tric power generating costs.
Without wishing to prejudge the question of payments on the fuel costs, we can nevertheless state that a con-
tribution to the correct solution of this problem will be made by an objective understanding of the effectiveness of
utilizing the circulating capital invested in the fuel inventory.
The values of the nuclear fuel on hand at a nuclear power station account for most of the power station cir-
culating capital. The circulating capital associated with the fuel inventory includes the value of fresh fuel (in the
form of hot channels, fuel assemblies, fuel clusters, fuel elements, etc.), and the value of the nuclear fuel after
drainage from the reactor and core reloading, to be regarded in fact as a semifinished product for further reprocessing
Translated from Atomnaya gnergiya, Vol. 20, No. 5, pp. 379-384, May, 1966. Original article submitted
January 29, 1966.
424
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(fuel recovery). This article considers only that portion of the power station operating capital which is tied up in
the fuel inventory.
At the present the accepted criterion) for the economic effectiveness of fixed capital is given by what we term
the calculated expenditures Z, defined as
z = C EnK,
(1)
where C is the net cost; En is the standard economic effectiveness factor of the fixed capital; K is the fixed capital.
An attempt to take into account that portion of the surplus labor corresponding to social expenditures made in
the production of the end commodity is undertaken by adding the EnK term to the net generating cost. The use of
the calculated expenditures category as ,a criterion for the economic effectiveness of nuclear fuel utilization leaves
room for a more correct evaluation of the costs savings achievable than the use of the net cost category alone.
It is generally acknowledged that one of the basic tendencies in current power reactor design is the constant
striving to increase nuclear fuel burnup, resulting in a lower fuel costs component CT as part of the total net gener-
ating costs; the fuel component is related to the average burnup ri as:
CT ?
Brj
'
(2)
where n is the net efficiency of the nuclear power stations; ZCi is the sum of the fuel cycle costs (planned profits
not taken into account).
As a rule, increased burnup of nuclear fuel realizable by way of improvements in the fabrication technology
of fuel elements and improved durability of fuel elements is brought about by increasing the initial fuel enrichment.
This, plus certain additional expenditure involved in routinizing the fabrication technology of improved fuel ele-
ments and in designing additional excess reactivity compensating facilities leads to a higher cost of the new fuel
charges, which can amount to rather considerable sums in some instances. This raises the question of whether any
and every lowering of the fuel component in the electric power generating costs at a nuclear power station will prove
economically effective. Currently available procedures [1] indicate that lowering net generating costs (in the spe-
cific case of the fuel component of the electric power generating costs) is economically effective only in the case
where the size of the cost drop ACT obeys the constraint
ACT> EnAKfi, (3)
where AKfi is additional operating capital invested in the fuel inventory.
It would be ill advised to base the approach solely on lowering the fuel component of the generating costs at
a nuclear power station, since in that case the monetary expression of the fuel component would end up expressing
nothing more than specific expenditures in means of production and in the necessary labor expended in the fuel cy-
cle, and would not reflect labor expenditures in the production of the surplus product for social use. A part of the
surplus product previously made available to society and directed to the production of new surplus product constitutes
productive capital, as we known from economic theory, but in this concrete case it refers to the additional circulat-
ing capital invested in the fuel inventory. In this instance the standard economic effectiveness factor En expresses
the ratio of the surplus product involved in the use of nuclear fuel to the value of the surplus product, at the given
moment. In other words, the economic category of the fuel component of calculated expenditures on electric power
generation [2] takes into account the addition to the fuel component of the electric power generating costs of the
value of precisely that surplus product in the fuel cycle (expressed in money terms) which is formed per unit electric
power generated at the nuclear power station.
A par of the surplus product finds a real monetary expression, with the introduction of the concept of payment
of productive capital. Since the rate of payment for the use of productive capital will reflect only the value of a
part of the surplus product, the sum of the generating costs and rate of payment for the use of productive capital can-
not be regarded as a necessary and sufficient criterion of economic effectiveness.
Nevertheless there are some engineering costs problems in which this sum does constitute a necessary and suf-
ficient criterion of economic effectiveness. Here, we have in mind all optimization problems in which only the
basic productive capital and the circulating capital vary, other conditions being equal.
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Setting a scientifically justified payment rate for both fixed capital and circulating capital in a nuclear power
plant is a matter acquiring fundamental importance. In the view of the present authors, the rate of payment in the
nuclear power industry must be arrived at on the basis of two factors: the relative capital consumption in production
and the interchangeability of nuclear-fuel and fossil-fuel electric power generation. We know that nuclear electric
power has been called upon to replace "conventional" or fossil-fuel electric power primarily in those districts of the
USSR where this is most feasible, i.e., in the European sector of the USSR In addition, studies carried out in the
USSR indicate that the relative capital consumption of nuclear and fossil-fuel power generation (on the scale of sev-
eral tens of millions of kilowatts) is about on the same level in those districts [3]. This means that the rate of pay-
ment of fixed capital in the nuclear power industry must be the same as (or at least not below) the rate prevailing in
the fossil-fuel and hydroelectric sector. The high relative capital consumption of production typical of nuclear and
conventional power generation possibly calls for a rate of payment on fixed capital lower than that set for branches
of the national economy which eat up less capital. On the whole, however, it is worth noting that the question of
whether the capital payments can be written off on the basis of unit rates or differentiated rates requires further
discussion.
Of course, the rate of payment on fixed capital cannot be viewed as invariant in the nuclear power industry.
With growth in technical process and changes in payment on fixed capital, the rate of payment will have to be re-
viewed periodically, remaining constant over each planning period in nuclear power development. It is indicated
in the proceedings of the September (1965) Plenum of the Central Committee of the Communist Parr" of the Soviet
Union that payment on capital purports to be a crucial part of the state buaget revenues.
The rate of payment on circulating capital in the nuclear power industry is a question meriting special atten-
tion. The overwhelming bulk of the circulating capital in nuclear power is the value of the nuclear fuel on hand.
While the amount of operating capital tied up in fuel inventory in a modern coal-fired electric power generating
station is no greater than 1-2% of the fixed productive capital of the power station, in the case of nuclear power
stations the amount of operating capital tied up in fuel inventory can run to 20-40% of the power station fixed cap-
ital assets. In addition, for a number of obvious reasons the rate of turnover of capital tied up in conventional fuel,
or the rate of turnover of any other forms of operating capital, runs tens of times higher than the rate of turnover of
the working capital tied up in fuel procurement and fuel inventory for nuclear power generating stations.
This means that the circulating capital tied up in fuel inventory for nuclear power stations shares certain com-
mon features with basic fixed capital, both in the relative size and in rate of turnover. These features must govern
the approach to setting the standard rate of payment for the use of nuclear fuel. In the view of the present authors,
the rate of payment must be set close to the rate of payment on fixed capital in the nuclear power industry, and in
any case must be not lower than the rate of payment for conventional circulating assets.
It is clear that the payment on the use of nuclear fuel must be collected during the time the fuel is still in
the fuel cycle. The rate of payment must be assumed unitary for all links in the fuel cycle over a sufficiently long
time interval (not less than three to five complete periods of the fuel cycle, or about two decades). Hence, the in-
troduction of a unit rate of payment for the use of nuclear fuel covering an entire branch of the national economy
(those enterprises involved in the fuel cycle) will provide an incentive for the routinization and use of those tech-
nological methods of nuclear fuel reprocessing which provide the least residence time of the fuel in all links of the
fuel cycle (from the standpoint of hitting upon a method of spent fuel recovery which would be preferable, for ex-
ample a method requiring no previous protracted radiation cool-down of the spent fuel elements). In a formal sense,
we could set a unit rate of payment for the use of nuclear fuel based on the assumption that the circulating capital
(price of the fuel inventory) is paid out over the fuel cycle duration, and since the terminal useful product of the
nuclear power industry is electric power, payment for the use of nuclear fuel over the entire fuel cycle would be in-
cluded in the cost of electric power delivered by the nuclear power generating station.
We see readily that if this payment is to be collected solely from the nuclear power station, all the remaining
enterprises involved in the fuel cycle will be indifferent to accelerated conversion of the nuclear fuel in the fuel
cycle. Further, in that case, uneven consumer costs in the fuel inventory will be levelled out in the various links of
the fuel cycle. Despite the unequal amounts of valuable products embodied in a unit of spent fuel, and the cost of
extracting them, the payment for the time of residence of the fuel in the fuel cycle will be determined by the cost
of fresh fuel, and this is wrong. It obviously makes no difference for society as a whole whether the cost of the spent
fuel is realized or not. In other words, for the state as a whole any nuclear power station is in essence always seen
as at least a dual-purpose utility. The cost of the nuclear fuel varies as it moves through the fuel cycle. As the nu-
clear raw materials are converted to fuel elements (reactor channels), the cost of reprocessing will be added to the
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cost of the nuclear raw materials in all links of the fuel cycle. It is typical of the utilization of fuel in nuclear
power stations that the spent fuel always has a certain finite value and, in some instances(this should be stressed), this
value can be higher than the initial value, or higher than the value of fresh fuel,*
If the average value of the nuclear fuel in the i-th link in the fuel cycle is denoted as "Ci (in rubles per kg U),
then the total specific charge for use of the fuel (during the fuel cycle) can be expressed as
P =
iTi rubles/kg U, (4)
where Pt is the standard effectiveness factor for circulating capital tied up in nuclear fuel, or the payment rate for
the use of nuclear fuel, in units per year; Tt is the holdup time of the fuel in the i-th link of the fuel cycle.
The unique rate of payment for the use of nuclear fuel in all links of the fuel cycle is a precondition, in the
view of the present authors, for economic stimulation of the nuclear power industry. The optimum performance of
individual enterprises involved in the fuel cycle in the nuclear power industry must be determined by the optimum
performance of the fuel cycle taken as a whole (the optimum performance of the entire branch of industry). The
introduction of a single payment rate for the use of nuclear fuel in all links of the fuel cycle is, therefore, a precon-
dition for economic incentives to speed up the movement of fuel through the fuel cycle. Additional holdup of nu-
clear fuel, e.g., values at K rubles over a time of T years in any link in the fuel cycle, will have to be expressed
in a surcharge amounting to PtKT rubles.
If the amount of fuel on hand in the reactor is GT (kg U per reactor), then the total budget deductions for the
use of nuclear fuel by enterprises involved in the fuel cycle (during the fuel cycle) will be
P p=p GICiTi rubles/reactor.
t
The charges accruing to the nuclear power station in that sum come to
NPS = pt G('NpsTNps rubles/reactor.
P
(5)
(6)
This raises the questionofhow budget deductions should be related to a given level of circulating assets tied
up in nuclear fuel, i.e., what should be considered the average value .ffNps of the nuclear power station fuel inven-
tory over the in-pile residence time of the fuel.
Since only the initial and final states of the nuclear fuel are realized from the standpoint of consumer value,
the average value of the fuel inventory during the in-pile residence time must be taken as the arithmetic-mean
value of its initial and final values;
1 ?-,
NAS? (- ft. p f )rubles/kg U,
s.
(7)
where Cfr. f is the value of the fresh fuel in the form of fuel.elements (hot channels, in rubles per kg U); Csp j is
the value of the spent fuel, or the value of the fuel remaining or accumulated after holdup charges, transportation
charges, and nuclear fuel reprocessing charges have been deducted, and with losses due to freezing of assets in these
links of the fuel cycle, in rubles per kg U, taken into account. Csp. 0 of course in every case, since if the hold-
up, transportation, and spent-fuel recovery charges were greater than the cost of the extracted and reprocessed fuel,
then fuel recovery would be economically unfeasible.
This means that the result of the cost calculations is independent of the variation pattern of cost indices of
the fuel inventory during the reactor run, which is to be determined by the special features of the processes taking
place in the reactor. Only the initial and final values of the fuel inventory cost are important.
*E. g., fuel located in the shields of fast reactors.
f Added to the necessary reserves of fuel stocks on hand at the nuclear power generating station.
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In particular cases where there is complete burnup of the nuclear fuel (if the innumerable value fission frag-
ments or radioactive isotopes are neglected) or where there is no recovery of spent fuel (the consumer value of the
spent fuel will be zero in that case)
Nps 'Cfr.f rubles/kg U.
Since the time the fuel is in the reactor is related to the reactor run by the formula
TN.ps = T?Iy,
the nuclear power station payment for the use of nuclear fuel is given by the formula
1
Pt == 2 Pr' (Cfr Csp. f )(7,7 .7' rubles /reactor.
(8)
(9)
The variable Pr characterizes that part of the surplus product created by the nuclear power station labor force
and associated with the use of the power station nuclear fuel inventory over a period of Tk/cp years.
The value (cost) of the electric power generated by the nuclear power station during the run must therefore in-
clude payment for the use of circulating capital tied up in the fuel, to the extent indicated by formula (9). The size
of the specific payment (payment per kilowatt-hour of electric power delivered to consumers) is
100p t Cfr f +-Csp
2 f kopecks/kW. h.
Pt 8760Veff
(10)
where Jeff is the average specific fuel irradiation at the power station, including the required fresh fuel reserves, or
the effective average specific irradiation of the fuel, in kW(th) per kg U.
If reserves of fresh fuel sufficient to fuel the power station on full power for tres years are on hand at the power
generating station, then
f eff= (1
(Pt res
T,
where J is the average specific irradiation of the fuel present in the reactor.*
We realize from formula (10), that the size of the payments must be inversely proportional to the utilization
factor of the nuclear power station installed plant capacity, the effective average specific irradiation of the nuclear
fuel, the thermodynamic efficiency of the nuclear power station, and directly proportional to the average value of
the fuel present at the nuclear power station, when we introduce the concept of payability of circulating assets.
Remember that the size of the fuel component in the electric power generating costs figured for the nuclear
power station is expressed in the general case by the formula
C =
?
100 Cfr.j ?Csp.f
24 kopecks/kW - h.
(12)
i.e., in contrast to the payment for the use of nuclear fuel, the fuel component in the electric power generating
costs is inversely proportional not to the effective fuel irradiation (10), but to the fuel burnup. The difference in the
values of the fresh fuel and spent fuel is transferred to the electric power value in this case.
The sum of the fuel component in the electric power generating costs and the payment for the use of circulat-
ing capital tied up in nuclear fuel inventory is given, hence, by the formula
*In this case we ignore any spent fuel present at the power station, since the spent-fuel storage basin built for spent
fuel holdup can be viewed in principle as an independent enterprise participating in the fuel cycle in its own right,
sharing the same premises as the nuclear power generating station proper.
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100 Cfr.f ?Csp.f
= 24 ?
B1
100pt Ch..f +Csnj
r
8760cpJeff 1 2
which reduces to
= 100 r Lr,
24Wri -fr.f )
'rfir f CVf
(1) res) 2
(13)
(14)
where 5 is the average fuel burnup, in kW(th)days/kg U.
The most efficient (optimum) utilization of the nuclear fuel corresponds to the minimized CI', in problems
in which variations in the payroll and fixed productive capital can be safely neglected. The optimum utilization
of nuclear fuel is achieved when the independent nuclear power station parameters are optimized, in which case the
equations
ac;
=0 (i = 1 , 2, ... , n) ,
a Xi
(15)
where n is the number of independent nuclear power station performance parameters, are satisfied.
It is important to emphasize the fact that conditions (15) are valid (to a sufficient degree of exactness in most
cases) only when a variation of the core parameters does not entail any variation in other performance characteristics
of the nuclear power station (and, hence, does not entail any further expenditures).
In the most general case (this applies to problems in which payroll variations can be neglected), optimality
conditions for the independent parameters can be stated in the form of a system of equations
ac*
axi
=0 (i = 1, 2, ... , n), (16)
where C* = C + P; C is the generating cost of the electric power delivered to the consumers; P is the payment for
the use of the nuclear power station fixed capital and circulating capital.
In the most general case, finally, the optimality condition of the independent power station performance pa-
rameters is stated in the form of a system of equations
aPw-0 (i?i, 2, ... , n),
axi
(17)
where Pw is the wholesale price of the electric power generated.
It is readily seen that conditions (15) and (16) are particular cases of the general condition (17), since the
wholesale price is made up of the generating costs and the surplus product, one part of which is calculated as pro-
portional to the payroll, the other part as proportional to the productive capital,.
Strictly speaking, as a consequence of the fact that the basic or independent performance parameters of nu-
clear power generating stations are quasi-independent, practical optimization of the cost indices of nuclear power
(t"
stations is an exceedingly complicated and cumbersome undertaking, but nevertheless a crucially important one,
the solution of which could yield important benefits to the national economy. The development of optimization
techniques, one of the most important problems in nuclear power work, is particularly needed under present condi-
tions where transitions to new techniques of economic management are in progress.
The introduction of payability of productive capital is an important economic incentive for more rational use
of nuclear fuel in the fuel cycle, and smoothes the way for a new and more profound review of fuel cycle economics.
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SucTh important questions as optimum time of introduction and productivity of enterprises participating in the fuel
cycle can be solved with economic benefits only by taking the payability of productive capital into account. On
comparing various types of reactors, we see that the circulating capital tied up in the formation of secondary nu-
clear fuel takes on the role of a factor displacing capital investments into the mining industry and to a partial ex-
tent into uranium reprocessing plants incorporated in the fuel cycle. Hence, the mathematical tools for investigating
the comparative economic effectiveness different reactor types can include concepts of negative capital investments,
and the introduction of the payability concept makes it possible to gain a correct understanding of the .economic sig-
nificance of the rate of turnover of recovered fuel.
The present article of course makes no pretense to an exhaustive coverage of the entire scope of topics arising
in nuclear power economics with the introduction of payability of fixed productive capital and circulating assets.
The utilization of this economic incentive, combined with others such as profit incentives which take into account
some natural conditions (alienation of territories for nuclear power station construction sites and for other fuel-cycle
operations), .can bring about a change in concepts held on the economic indices of nuclear power, and will, there-
fore, require further study.
LITERATURE CITED
1. Basic Techniques for Power Engineering Costs Calculations [in Russian], Moscow, State Tech. Lit. Press (1959).
2. Yu. I. Koryakin, V. V. 13atov, and V. G. Smirnov, Atomnaya energiya, 17, 94 (1964).
3. N. M. Sinev, B. B. Baturov, and V. M. Shmelev, Third Geneva International Conference on the Peaceful Uses
of Atomic Energy, Paper No. 294 (1964).
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HEAVY-CURRENT ACCELERATOR BASED ON A TRANSFORMER
E. A. Abramyan and V. A. Gaponov UDC 621.384.60
The operating principles of a direct-action accelerator designed to accelerate electrons to an energy
of 1.5 MeV with a mean beam power of tens of kilowatts and an efficiency of around 90% are de-
scribed. The electron?current pulse length can be varied from 0 to 5 msec, and the repetition fre-
quency up to 50 times per sec. The mean current im may reach 1/6 of the maximum current in the
pulse. Magnetic lenses are installed in order to focus electron currents of up to 100 mA into a beam
a few mm in diameter in the accelerating tube. Heavy-metal screens are placed close to the axis
of the tube in order to protect the gas gaps and other electrically-stressed parts of the accelerator
from radiation arising inside the tube.
The construction of a system for producing an electron beam with an energy of 1.5 MeV and
a mean power of 25 kW (im = 17 mA) is described.
Accelerators producing 1 MeV electrons and upward find wide application in both physical investigations and
various applied systems. Accelerators are used as radiation sources in chemistry, for the sterilization of medical
appliances, and for the conservation of products. Electron beams with energy densities of tens of kW/cm2 enable
melting, welding, and other metal-treatment processes to be carried out. For energies of many hundreds of keV,
the electrons can be brought out of the vacuum chamber in which acceleration takes place and used in air or an
inert-gas medium at atmospheric pressure or higher.
In 1963, on the initiative of G. I. Budker, in the Institute of Nuclear Physics of the Siberian Branch of the
Academy of Sciences, USSR, work was begun on setting up a high-efficiency, high beam-power accelerator for ap-
plied purposes. Of the many known and possible means of accelerating charged particles, acceleration by means of
a transformer [1] was selected. In 1964, the first experimental LT-1.5 apparatus was constructed and prepared, this
being an electron transformer for accelerating electrons to energies of 1.5 MeV. We here present the main compo-
nents of the apparatus and its operating principles.
Arrangement of the Apparatus
The main constructional arrangement of the apparatus is shown in Fig. 1. The primary winding of the trans-
former, 1, and a section of the secondary winding, 2, are placed coaxially. The central part of the magnetic circuit
consists of individual, mutually-insulated discs 3, and is completed on the high-voltage side by the head 4. The
magnetic flux also passes through the outer components 5, 6, 14 of the magnetic circuit and through the base 16.
Sections 2 are connected in series (every two .sections fixed to a disc 3), and the middle point of two neighboring
sections is electrically connected to the disc. The accelerating tube 8 is mounted on the transformer and its elec-
trodes are connected to the discs 3. The injector 10 has a control electrode 9, which enables the electron current
through the tube to be varied as far as complete cutoff. Heating of the cathode and the voltage on the control elec-
trode are provided by the injector supply system 11, fed from coil 7, which forms part of the secondary winding.
Capacity divider 13 serves to regulate the voltage on electrode 9. In addition to the control system, the head 4
carries a condenser battery 12 connected to coil 7 (the purpose of the condensers will be explained in a moment).
The whole transformer is placed in the vessel 15, filled with compressed gas. The vacuum pump 17 ensures a vac-
uum of 10-5 to 10-6 mm Hg in the tube. For scanning the beam and releasing electrons into the atmosphere, we
have a magnet 18 and a "trumpet" 19 carrying a foil. The dimensions and position of the beam are determined by
probes 21, and the current is measured with a screened Rogowski belt 20.
Operating Principles and Characteristics of the Transformer
Figure 2 shows the equivalent circuit of the transformer, where L is the inductance of the transformer, Lis is
the stray inductance of the primary winding, C is the total capacity of the transformer, and RL is the load resistance.
Translated from Atomnaya gnergiya, Vol. 20, No. 5, pp. 385-392, May, 1966. Original article submitted
January 25, 1966.
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15
14
13
12
11
10
9
32 33 34 20 1f
31 30 29 28 27 26
Fig. 1. Construction of the apparatus. 1) Primary winding of the tans-
former; 2) section of secondary winding; 3) disc of magnetic circuit;
(head); 5, 6, 14) components of magnetic circuit; 7) head coil); 8) ac-
celerator tube; 9) control electrode; 10) injector; 11) injector supply
system; 12) condenser battery; 13) capacitive element of head; 15) ves-
sel; 16) principal magnetic circuit; 17) vacuum pump; 18) rotating
magnet; 19) "trumpet" with outlet window; 20) Rogowski belt; 21) probe;
22) copper rings; 23) support cylinders; 24) elastic contacts of accelerator
tube; 25) primary-winding screen; 26) screen for section of secondary
winding; 27) protective discharge tubes; 28) ohmic divider; 29) magnetic
lenses; 30) radiation screens; 31) electrodes of accelerating tube; 32) ball-
bearing support for tube; 33) tube sealing; 34) radiation shield; 35) cooling
radiator; 36) ventilator.
Fig. 2. Simplified transformer circuit: L = trans-
former inductance; Lis = stray inductance of pri-
mary winding; C = capacity of transformer,
RL = load resistance.
432*
Let us consider the case in which the power lost in the transfor-
mer is much smaller than the useful power P. Let us also sup-
pose that the natural frequency of the LC circuit equals the fre-
quency of variation of the voltage tri applied to the primary
winding. As already indicated, thanks to the presence of a con-
trol electrode on the injector, the current through the tube can
be zero even in those half-periods in which the polarity of the
voltage on the secondary winding corresponds to acceleration of
the electrons (minus on the high-voltage end of the tube). While
there is no current in the tube, the transformer operates under
no-load conditions and the current i1 in the primary winding is
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utube
itube
Fig. 3. Voltage and current variation in tube: utube = voltage on accel-
erating tube;tube = current of accelerated electrons.
i
small. Let us suppose that the control electrode begins to open at a moment t1 (Fig. 3). It is easily seen that, for
a particular law of variation of the current itube in the accelerating tube, the voltage utube on the latter may re-
main constant. Let us call wI and wll the number of turns in the primary and secondary windings and k
the transformation factor; then for the time interval t1 < t < t2 we may write: ulm sinwt = LIs(dii)/dt + kutube. In-
tegrating bothsides of the equation, putting utube = const, and considering that uim/k = u/im, we obtain
11 utubeLkis [ (cos.ti_cos.,t)_(t _to .
?
be
(1)
During the time t1 < t < t2, the voltage in the inductance L is also constant L(diL)/dt = utube, whence, the current
in the inductance is iL' /L(t?ti) + iL (t1). The value of Oh) can be found by considering the no-load utube con-
dition:
"
L =uin, sin cot; i (t1)= um
cos aiti.
dt coL
For constant voltage on the capacity C, the current ic = 0, and the current in the tube itube = kii?iL. Denoting
= sin = a and Lis/L = v, we obtain
utubeiullm
//TT
i ? " m
Lis [(k2+ v) cos cot
tube co
? aft) (k2 ? v) (t ? t5)? k2 cos WI.
For itube = 0, at the moment t2, Eq. (2) takes the form
(k2 v) cos (di ?ao) (k2+ v) (t2 ?
k2 cos c0t2 = 0.
We may find t2 from Eq. (3).
t2
The mean power in the beam P = futuJ
?2
"" tube
P = -
rtubedr, or
[(k2 v) (t2? t1) cos a0t1
(2)
(3)
(4)
au.) k2
(k2+ v) (t2? t1)2? ?co (sin ot2? sin WI)
2
The formula obtained in P is approximate and reflects the qualitative dependence of useful power on the pulse
length h?t, the stray inductance LB, and the ratio v = Lis/L. A more exact relationship between P and all the
parameters in the steady-state condition, allowing for the distributed parameters of the transformer, was found from
models. For the apparatus described in this paper P = 25 kW for t2?t1 = 5 msec, LTc = 4.6.10-4 H, L = 40,000 H
and v.1.15108.
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The method here considered for keeping a constant voltage on the tube by controlling the tube current may
also be used in order to produce any other desired form of voltage, utubc(t), by due choice of the law governing the
variation of itube during the pulse.
As indicated earlier, the natural frequency of the circuit was chosen to be equal to that of the voltage supply-
ing the transformer. Thus, for the apparatus here described f - /-- - 50 cps, and the supply for the primary wind-
2Try LC
ing is taken directly from the mains.
In order to ensure constancy of the electric-potential gradient along the central part of the magnetic circuit,
the number of ampere turns in each air (or more strictly gas) gap should be proportional to the corresponding mag-
/ i
netic resistance R ( Ri = -- ----, There are two sections 2 in each gap 6 between the discs (see Fig. 1). The gap l"
be-
tween the highest disc and the head 4 is half as wide and, therefore, contains only one section 2. The same holds
for the gap between the lowest disc and the base of the magnetic circuit 16. The ratio of the number of ampere
turns in coil 7 to the total number in all the sections equals Ra /16R6, where Ra is the magnetic resistance of sec-
tion a to which the whole voltage is applied, and Ro is the magnetic resistance of the gap 6 between discs. The
equivalent circuit of the magnetic system with the secondary winding is shown in Fig. 4, where 6' and e are the
number of ampere turns on the coil 7 and section 2 respectively. On the basis of the condition that there should be
approximately the same electric field in the gas gaps, the value of a equals the sum of all the interdisc gaps (or a
little more than this), so that Ra ? R6. The peak value of the ampere turns in coil 7 is i7mw7 = .t.mRa, where .t,m
is the amplitude of the principal flux. In order to ensure the necessary calue of i7mw7 for a relatively small voltage,
a current several orders larger than that passing through sections 2 is passed through the turns of coil 7. In order to
obtain the desired value of i7m, a condenser battery 12 is connected to coil 7; the capacity of the battery is deter-
mined from the relation C12 -- where u7mW7
= Umn-. On satisfying these conditions, the amplitudes of the
cou7m, wII
magnetic energy in the gap a and the electric energy in the capacity 12 will be equal, which follows from the iden-
tity existing between the conditions i7mw7 = mRa, 7rn wC12um on the one hand and Co.& = 4>2mRa on the other.
One of the peculiarities of the transformer described is the large number of gaps in the magnetic circuit. It
is easy to show that this arrangement will have a maximum Q factor for a certain ratio 6/A, where A is the height
of the discs 3. As we know, the Q factor is determined from the relation Q = Pr/(Pe + Ps), where Pr is the reactive
power of the transformer, and Pc and Ps are the copper and steel losses respectively. If the operating frequency, the
magnetic-circuit material and the material of the windings, and the dimensions of the magnetic circuit (over-all
length and cross section at each point) are given, then
.2
L
Pr = ki R '
.2
7. IL
k2i1 and Ps ?Rti 7
/i
where R = is the total magnetic resistance (reluctance), and k1, k2, and ks are coefficients constant under
rik.)
the given conditions. It follows from this that
n kiR
k2R?k3
(5)
We see from Eq. (5), that Q is a function of the magnetic resistance of the circuit, R. From the condition dQ/dR =0
we find the value of R* =
giving
VT:cl
k2
R* = Tirc3
piSi 7172
corresponding to maximum Q factor of the transformer. The values of 6 and A,
easy to calculate. It is not difficult to see that for R = R* we have Pc = Ps,
i.e., when the losses in the windings and in the steel of the transformer are equal the Q factor reaches its maximum.
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Thus, in the apparatus here described the size of the gaps 6 between the discs is determined by the electric
strength required and by the condition corresponding to maximum Q factor.
Parameters of the ElT -1.5
Before describing the structural characteristics of the glT -1.5 accelerator, let us give its principal parameters:
Supply voltage of primary winding
Maximum voltage on secondary winding for no-load running
Maximum accelerating voltage on tube
Range of variation of accelerating voltage
Stability of accelerating voltage
Mean power of electron beam in the main operation
condition (utube = 1.5 mV)
Range of variation of mean power
Value of a for the principal operating condition
Current pulse length for the principal condition
Mean current for the principal condition
Maximum current in the pulse for the principal condition
220/380 V. 50 cps
1.7 mV
1.5 mV
400 kV to 1.5 mV
up to 2%
Total losses in the windings and magnetic circuit of the transformer,
the injector, and its supply system for the principal condition
Transformer inductance L
Total transformer capacity C
Natural frequency of the. LC circuit
Stray inductance of the primary windings in the principal conditions
Dimensions of accelerator with outlet device and vacuum pump
Dimensions of containing vessel
Gas pressure in vessel
Weight of accelerator
Construction of the 1T-1.5
25 kW
0 to 25 kW
0.88
5 msec
? 17 mA
100 mA
? 2.5 kW
4 ? 104 H
250.10-12F
50 cps
4.6. 10-4 H
height 3.3 m
width 1.3 m
height 2.1 m
diam. 1.2 m
15 atm
8 tons
Transformer. The magnetic circuit of the transformer is made of g43 transformer-steel plates 0.35 mm thick
cemented with epoxy resin. The discs 3 (16 of these in all), the head 4, and other components of the magnetic cir-
cuit are formed of plates arranged radially; after applying the resin, the end surfaces are specially treated to give a
good finish. The voltage on the gaps 6 (in our apparatus 6 = 6 mm) is around 100 kV for a nominalvoltage of 1.5 my
on the accelerator. The gap a = 120 mm; the maximum field in the gas is about 160 kV/cm for a voltage of 1.5
mV on the secondary winding. The head 4 (Fig. 5) and other, components of the magnetic circuit are treated after
cementing in order to obtain the desired shape and finish of the surface. In order to avoid short-circuiting the plates
in the course of machining, the insulation between the plates is made up to 0.15 mm thick. The discs and head
have central openings with built-in closed copper rings 22 (see Fig. 1). The bearing insulating cylinders 23 are sup-
ported on the rings. The transformer-iron places of the discs
are electrically connected to the rings 22, which almost en-
tirely screens out the magnetic field of the transformer in
the tube region.
Ra
The primary winding 1 is mounted on a copper screen
25; glass ribbon and epoxy resin serve for insulation. Each
section 2 of the secondary winding consists of 3550 turns of
copper conductor 0.1 mm2 in cross section. The sections are
connected in series. The operating voltage on each section
is about 50 kV. We see from Fig. 1 that the gap between
neighboring sections 2 varies with radius, thus, ensuring an
approximately uniform electric field between sections. The
value of the test voltage on the sections is greater than the
operating voltage by a factor of 1.5 to 2. The screens 26
protect the secondary winding from damage in electrical
breakdown, and the discharge tubes 27 from overvoltage on
J.
Fig. 4. Equivalent electrical
circuit of the magnetic circuit;
6 ampere turns of the head coil;
e ampere turns of the section of
secondary winding; Ra magnet-
ic resistance between the head
and outer components of the
magnetic circuit; R5 magnetic
resistance of the gap between
the discs.
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the sections in the course of conditioning. The layers of the sections
are insulated with paper impregnated with epoxy resin. After a
special impregnation process, the sections become monolithic with
the necessary electrical and mechanical characteristics. Coil 7has
36 turns. The condenser battery 12 consists of 24 condensers with
a total capacity of 480 I.LF and an operating voltage of 500 V; they
are set in a space of the head 4. The transformer dimensions and
the number of secondary windings chosen given the system a natu-
ral frequency of 50 cps.
Injector and Accelerating Tube. The injector of the experi-
mental apparatus has an indirectly-heated lanthanum-boride cath-
ode 5 mm in diameter and a control electrode 9. For a voltage of
1.5 mV on the whole tube, the voltage relative to the cathode giv-
ing complete current cutoff when applied to the control electrode
is 2 kV.
Fig. 5. Central column of transformer.
The accelerating tube is demountable and consists of eight
identical sections interconnected by rubber seals. Each section is
made of epoxy rings cemented to Dural electrodes. The total length
of the tube is about 1.2 m and the number of insulating rings is 64.
The'voltage is taken to the electrodes at 18 points by means of
elastic contacts 24 connecting the electrodes to the discs 3, the
head 4, and the base of the magnetic circuit 16. The voltage for
the intermediate rings is conveyed by an ohmic divider 28. At the
present time ceramic tubes having a minimum number of vacuum joints are being tested. The tube is provided with
16 short-focus permanent magnetic lenses 29; these give a beam diameter of 3-5 mm for a current of around 100 mA.
Screens 30 made of an 88/0 Pb alloy are placed at the tube axis. These substantially reduce the radiation
arising when electrons fall on the electrodes and so forth. To facilitate evacuation, the electrodes 31 contain aper-
tures displaced relative to each other in neighboring electrodes. The tube is evacuated with an N5S pump using a
nitrogen trap.
The injector and tube may be removed through the head 4 without taking the vessel away. The electrodes
31, together with lenses 29 and screens 30, are easily taken out and may be replaced for experimental purposes or
when any component goes out of order during use.
The ball bearing 32 and tube sealing 33 ensure smooth fixing and the absence of stress if the insulator in the
tube is not coaxial with the magnetic-circuit column.
Containing Vessel, Outlet Window, and Other Systems. The vessel 15 in which the transformer is sited is filled
with a mixture of freon and nitrogen at a pressure of up to 15 atm. The gas passes into the vessel through a drying
chamber. After overhauling, the whole accelerator is dried under vacuum.
The construction of the system designed for bringing the electron beam out into the atmosphere is determined
by the conditions governing the specific technological process in which the accelerated electrons are to be used. In
the experimental accelerator, the electrons are let out through a window of titanium foil 50 ji thick and 400 x 40
mm in area, cooled with compressed air.
In order to prevent the radiation arising when the beam is retarded by the object being irradiated from falling
into the transformer, a steel shield 34 is placed under the vessel.
Despite the high efficiency of the accelerator, the losses in the windings and magnetic circuit of the trans-
former, together with those in the injector and its supply system, amount to around 2.5 kW. In order to eliminate
heat, there is a water-cooled radiator 35 and a ventilating blower 36, which ensures the necessary gas circulation
in the vessel.
Control and Automatic-Control Systems of the Accelerator. The principal control and automatic-control sys-
tems of the accelerator are shown in Fig. 6. The voltage on the secondary winding is regulated by varying the trans-
formation coefficient, for which purpose several tappings of the primary winding are provided, together with a switch.
The voltage on the secondary winding is measured in steps of 50-100 kV. Smooth regulation of the voltage on the
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r
2\
tube is effected by varying the voltage drop a = utube/uHrn as a re-
sult of changing the current-pulse width t2?t1 and, hence, slightly
changing the mean power P. Stabilization of the assigned power is
also effected by adjusting the width of the current pulses. A power-
control and stabilization circuit is provided for these operations; the
sensitive elements of this are a Rogowski belt and the capacitive
voltage control on the tube.
The injector supply system generates the voltage on the con-
trol electrode u9 needed to produce the appropriate itube(t) for main-
taining a constant voltage on the tube. If it is necessary to keep
utube constant to 2% or better, a feedback system is set up in the
injector supply; this is controlled by capacitive tube-voltage control
element situated in the head 4 and provides the u9. When it is per-
missible for the electron energy to vary by 5-10%, the control system
may be replaced by a source of constant emf, completely cutting off
the injector current, capacitive control on the head being connected
in opposition to this. Then the injector will be opened for a given
time t2?t1, depending on the relation between the steady emf and
emf arising on the capacitive control. It is also possible to have an
arrangement ensuring utube = const or some other utube(t) law in
accordance with an assigned program.
The injector supply system also generates heating for the cath-
ode. The system itself is fed from some of the turns of the secondary
winding. In order to ensure constant supply of the cathode and other
circuits situated in the head 4 when the tube voltage is varied be-
tween 400 kV and 1.5 mV, an appropriate stabilization system is pro-
Vided in the injector supply. The various devices for regulating the
parameters of the high-voltage components of the supply system(cath-
ode heating, current pulse width, etc.)are controlled, as in ordi-
nary electrostatic generators, by means of insulated filaments. A
light-operated feedback scheme for generating u9 has also been
successfully tried. The light source is placed in the grounded part and a photodiode in the head 4. With this sys-
tem, the moment ti at which the control device begins to operate and the current starts flowing through the tube is
given at each period of the voltage. In this case, the electrons pass through the tube either as single pulses or with
any assigned repetition frequency (up to 50 times per sec).
A time-base generator ensures a linear variation of field in the magnet scanning the electrons along the long
side (400 mm) of the outlet window; the scan frequency is 2 kc/sec. The scanning angle is kept constant for dif-
ferent electron energies by a time-base control and stabilization circuit. In the perpendicular direction the beam
is scanned by a sinusoidal voltage of industrial frequency.
A switch system is provided in order to switch off the line voltage for external operation of any of the accel-
erator components.
11
Fig. 6. Block diagram of the control and
automation system: 1) Switch; 2) capa-
citive indicators of tube voltage; 3) in-
jector feed system; 4) power stabilization
and control system; 5) sweep stabiliza-
tion and control system; 6) sweep gener-
ator; 7) rotating magnet; 8) foil temper-
ature gage; 9) from pressure gage in
boiler; 10) mains; 11) mains cutoff sys-
tem; 12) Rogowski belt; 13) vacuum
gage; 14) breakdown indicator.
EXPERIMENTAL RESULTS
The g1T-1.5 was tested under various operating conditions. The beam was accelerated both in single pulses
and with a repetition frequency of up to 50 cps. The main beam-current measurements were made with a Faraday
cylinder situated in air and with a Rogowski belt. The beam energy was determined by means of a capacitive pickup
situated in the gap a. The mean power was measured both by a thermal method and from the power required by the
transformer, allowing for the steel and copper losses.
The electrons were accelerated in the energy range 400 keV to 1.5 MeV. For an energy of 1.5 MeV the mean
power was around 25 mW, the mean current 17 mA, the maximum current in the pulse 100 mA for a pulse width of
5 msec, and the accelerator efficiency approximately 90%. The pulse width was between 0-5 msec. The beam
diameter at the outlet from the tube was around 5 mm for a mean power of 25 kW. The beam scanning was carried
out in two directions, through angles of ?2.5 and ?25?. The maximum mean power obtained under short-lived con-
ditions was about 30 kW.
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CONCLUSION
The construction and operation of the first experimental t1T-1.5 accelerator show that the arrangement pro-
posed is feasible and the calculations forming a basis for the construction are valid. The high efficiency and consid-
erable mean power should enable us to use analogous systems for various high-energy radiation processes in metal-
lurgy and other fields.
Direct-action accelerators such as the dynamitron, the insulated-core transformer, and the resonance trans-
former [2] are closest to the apparatus described as regards parameters. The supply of the dynamitron and insulated-
core transformer demands high-frequency converters (the former at hundreds of kc/sec and the latter of about
1 kc/sec). In both systems, the voltage rectification demands a large number of rectifying units, which complicates
the reliability aspect. Resonance transformers also operate above industrial frequency.
Systems with hf acceleration (linear accelerators and microtrons), although having greater limiting energies
than the systems just mentioned, have relatively lower efficiencies, and plainly cannot compete with the electron
transformer in the energy range 1-3 MeV in cases in which beams with a power of many kW are required.
The main difference of the ]1T-type systems includes the rectification and stabilization of the accelerating
voltage by means of the electron beam itself, and also the supply of the system direct from an industrial-frequency
network (50 or 60 cps). In addition, these systems make it possible to carry out acceleration with an assigned energy
distribution of the electrons, which is required, in particular, when it is desired to obtain a uniform dose over the
thickness of the material irradiated.
A failing of the electron-transformer accelerator is the fact that the pulse power in the beam is much greater
than the average power. This, however, does not prevent the use of such accelerators, since in high-energy processes
the necessary doses are collected over many hundreds of pulses, and this is practically equivalent to the action of a
continuous beam. The most serious difficulties may arise in the accelerator tube, which has to pass a pulse current
several times exceeding the mean value.
The construction developed for the tube, however, enables the necessary pulse power AO be passed. Another
failing of the system described is the fact that the back voltage on the tube exceeds the accelerating voltage by up
to 10 or 15%. This does not affect the overall electric strength of the system, however, since in the absence of a
beam and its associated radiation (although weakened by the special screens), the apparatus (tube and gas gaps) is
able to withstand greater voltages.
Further development of the electron transformer should make it possible to raise the energy of the particles
obtained in such systems as well as the mean power. In the future, it should clearly be possible to obtain an effici-
ency close to that of an ordinary transformer (95-98%). At the present time the construction of a 1.5-MeV acceler-
ator designed for serial manufacture is being planned.
In conclusion, we consider it our pleasant duty to thank our colleagues in the Institute of Nuclear Physics of
the Siberian Branch of the Academy of Sciences, USSR, who have taken an active part in constructing and setting
up the apparatus: engineers G. Krainov, V. Nikolaev, and I. Shalashov, mechanics V. Biryukova, G. Balykov,
M. Voronov, M. Gubin, Yu, Efremenkov, A. Kotachev, and M. Stepanov, and technician V. Kirov.
LITERATURE CITED
1. E. A. Abramyan and V. A. Gaponov, A High-Efficiency System for Accelerating Charged Particles [in Russian],
Author's certificate No. 906570 as from August 31, 1964.
2. M. Cleland and K. Morganstern, Nucleonics, No. 8, 52 (1960).
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EXPERIMENTAL DETERMINATION OF THE RADIATION
QUALITY FACTOR NEAR HIGH-ENERGY ACCELERATORS
V. N. Lebedev, M. Zel'chinskii, UDC 577.391
and M. I. Salatskaya
Experimental data characterizing the effective quality factor QF of multicomponent pulse radiation
in various parts of a 10-GeV synchrophasotron are presented. The measurements were made by the
recombination method. The value of QF varied from 3 to 11. The results of the measurements are
compared with values determined for other high-energy accelerators. Values of QF obtained in ex-
periments on particle beams in a 680-MeV synchrocyclotron are also given.
Modern high-energy proton accelerators give rise to strong secondary radiation with a very complex compo-
nent composition. The energy range of each component stretches from fractions of an electron volt to an energy
close to the maximum energy of the accelerated protons. This diversity of radiation, together with the pulsed char-
acter of the process, makes it extremely difficlt to determine the degree of radiation hazard. In such a situation it
is in practice only possible to study the components giving the greatest contribution to the dose in sites occupied by
personnel [1-3]. Moreover, in estimating the contribution of very high-energy nucleons, various simplifying assump-
tions only satisfied to a first approximation are generally used.
However, even in the case of the absolute validity of these assumptions, an additional error inevitably arises
because the relation between the high-energy neutron' flux and the dose, as recommended by the rules of [4], cannot
be regarded as strict.
Use of the well-known experimental relation between the density of linear stopping power and the quality
factor QF* [6, 7] may provide a way out of this. By means of this relation we can calculate the effective QF of any
radiation if we know the corresponding spectrum of the linear stopping power (LSP). Knowing the value of the QF,
it is easy to estimate the dose equivalent in berads (biological equivalent of radiation). The ISP spectrum in turn
may be determined by means of a tissue-equivalent proportional counter [Rossi, 8, 9]. An estimate of the mean
LSP can also be obtained by an analysis of the tracks in nuclear emulsions or on photographs from track chambers
[10, 11]. Any of these methods, however, is unsuitable for rapid and operational measurements under practical
conditions.
The recombination method recently proposed [12, 13] offers the possibility of overcoming some of these dif-
ficulties and determining the effective quality factor of unknown radiation simultaneously with absorbed-dose mea-
surements, without requiring an analysis of the ISP spectrum. This method was used for determining the quality
factor of mixed pulsed radiation near the high-energy accelerator of the United Institute of Nuclear Research.
Measurements were made with the help of a double tissue-equivalent recombination chamber, described in
detail in [14]. The ratio of the ion currents in two chambers (one of which operated in the saturation condition, and
the other of which operated under column-recombination conditions) served as a quantity proportional to the quality
factor. In the column-recombination condition, the number of recombining ions depends on the linear density of
ions in the particle tracks.
The results of the measurements, averaged over the whole volume of the chamber, correspond to an effective
tissue depth of 2-5 cm, which is due to the structural peculiarities of the tissue-equivalent chamber. The error
*The term "quality factor" of radiation is proposed by the International Commission on Radiation Units and Measure-
ments (ICRU) [5] for denoting the physically-measurable parameter of radiation characterizing this radiation from
the point of view of the expected biological effect, in contrast to the term 'relative biological effectiveness" (RBE)
which is now recommended for use in radiobiology only.
Translated from Atomnaya gnergiya, Vol. 20, No. 5, pp. 392-396, May, 1966. Original article submitted
October 21, 1965.
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Effective quality factor of mixed radiation; a) 570-keV pre-injector; b) 9-MeV linear accelerator; c) yoke
of synchrophasotron electromagnet; d) rectilinear gap; e) high-energy particle channels; f) operating targets;
g) experimental room. Under the stroke: values of QF from [3].
arising as a result of the fairly large (30 x 30 cm) dimensions of the chamber (due to inhomogeneity of the radiation
field, variations of the spectrum over the volume of the chamber, and so forth) was not taken into account.
The results of measuring the effective quality factor of the radiation at the most characteristic points inside
the building of the 10-GeV synchrophasotron are shown in the figure. As expected, the QF reaches a minimum value
( 3) at places directly adjacent to the open rectilinear gap of the accelerator. At these points, the main contribu-
tion to the dose comes from primary and secondary high-energy particles (relativistic protons and neutrons, mesons,
and the electron?photon component) having minimum specific stopping powers. On moving away from the open
parts of the vacuum chamber, the quality factor increases sharply, and in the center of the accelerator room reaches
a value of 5 (mean energy of fast neutrons at this point equals 0.6 MeV [3]). Opposite to the window in the yoke of
the electromagnet (the yoke consists of four quadrants as shown in the figure, each containing 12 oval windows
(1 x 1.5 m) and 12 vertical slits 0.2 m broad) the quality factor is 3.8, and opposite the slits it is 4.5, which is due
to the different contributions of the high-energy component. The quality factor rises a very little beyond the 60-cm
thick wall of silicate brick. This is apparently because the fast-neutron flux, which falls off on passing through the
wall, is made up as a result of cascade processes. Beyond the thick shielding in the experimental rooms, the quality
factor rises to a maximum value of 6-10. This value of QF agrees closely with the mean neutron energy in these
regions (0.4-0.8 MeV) if we remember that in a small experimental room the high-energy component is practically
absent.
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TABLE 1. Effective Quality Factor of Radiation, Determined for Various Accelerators
Characteristics of regions
Accelerator
United Institute of Nuclear
Research
3-GeV
cosmotron
6.2-GeV
bevatron
recombination
method (see
figure)
estimate based
on composition
of radiation
In immediate proximity
(2-3 m) to the un-
screened vacuum cham-
ber of the accelerator
3
Outer and inner annular
zones along the yoke
of the electromagnet
(including the center
of the room)
3.4-5
4.1-6.5 [3]
1.8 ?20% [9]
Beyond the thin shield
(up to 60 cm) (site
where personnel may
be present)
5-6
4.5 [3]
5 [10]; 8.1 [11]
2. 8 ? 20% [9]
Beyond the thick shield
(in the plane of the
equilibrium orbit)
(site where personnel
are constantly present)
6-10
8.8-9.2 [3]
*At various points.
CERN 28-GeV proton
synchroton
1 4.3-4.1-6.0-5.9*
[16]
The measuring error is approximately 30%; it is due to the following factors: primarily to the incomplete
tissue-equivalence of the material composing the chamber, the incomplete saturation in the chamber operating in
the saturation condition (for strongly ionizing particles), errors arising on measuring the ion current of the chamber,
and a certain nonlinearity of the characteristics of the chamber operating in the recombination condition [15, 16].
However, the error in relative measurements (scatter in the readings of the same apparatus) is considerably smaller,
and, as might be expected, does not exceed 15%. This fact enables us to distinguish several characteristic regions
in the accelerator room as a function of the value of QF (Table 1).
The values given in the figure and in Table 1 agree closely, within the limits of experimental error (except
for one point), with the results of estimates based on some quite pessimistic views on the spectrum and composition
of the radiation [3].
The value of the effective quality factor of the radiation close to other high-energy accelerators has been
estimated by many authors [9-11, 16]. It is very hard, however, to make a comparison of the various results, since
the geometry of the shielding differs for different accelerators, and also the disposition of the measuring points with
respect to the accelerators is not always very clearly indicated. Nevertheless, we have tried to correlate the data
in Table 1. This table shows that the measured results agree fairly well with the estimates, except for the data re-
lating to the bevatron and obtained from an analysis of the LSP spectrum, i.e., the most reliable as regards the
method of obtaining information. It is extremely doubtful that these data were erroneous; however, there are no
grounds for ascribing a value of QF = 1.8-2.8 to the radiation in regions remote froni the accelerator (for example,
the control desk), which is very rich in the soft component [17]. We may suppose that this is due to the presence of
a large number of relativistic particles at the measuring points, although there is no direct indication of this in [9].
The results of measuring the quality factor of the radiation in collimated particle beams from the 680-MeV
synchrophasotron of the United Institute of Nuclear Research are presented in Table 2.
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TABLE 2. Comparison of the Effective Quality Factors of Radiation for High-Energy Particles
Type of radiation
Quality factor
data of this paper
published data
RBE
Neutrons, En max = 680 MeV
Protons, E = 680 MeV
Scattered radiation in experimental rooms
beyond thick shield
2.7?0.8
1.8?0.6
For 100-400-MeV protons.
tFor 510-730-MeV protons (data relate to live irradiations).
$At various points.
1.4* [18]
5.3?0.5-[9];
10-13-3.6-4 [16]
0.7?f [19-23]
For a beam of neutrons with a maximum energy of 680 MeV [24], the measurements were made in a water
phantom(model)[25]. The maximum of the absorbed dose corresponded to 24 cm of water. The change in the
quality factor with depth was insignificant and did not exceed measuring error, right up to 1.2 m. The quality factor
was measured in a 680-MeV proton beam without the phantom.
In the third and fourth columns of Table 2 we present, for comparison, some calculated or measured QF values
of RBE obtained from live irradiations of animals (mice and dogs).
In conclusion, the authors use this opportunity to thank V. G. Buyanin and E. I. Ob'ezdnov for making the
measurements.
LITERATURE CITED
1. R. Wallace, et al., Collection of Articles from the Symposium on Particular Questions of Dosimetry [Russian
translation], Gosatomizdat, Moscow (1962), p. 175.,
2. B. Moyer, Conference on Shielding of High-Energy Accelerators, New York (1957), p. 38.
3. L. S. Zolin, V. N. Lebedev, and M. I. Salatskaya, Preprint OIYaI No. 2251, Dubna (1965).
4. Health Rules for Working with Radioactive Substances and Sources of Ionizing Radiations, No. 333-60
[in Russian], Gosatomizdat, Moscow (1960).
5. "Izmerit. teldmika," 10, 54 (1963).
6. Recommendations of the International Commission on Radiation Shielding [Russian translation],
IL, Moscow (1958).
7. Health Phys., 9, 357 (1963).
8. H. Rossi and W. Rosenzweig, Radiation Res., 10, 532 (1959).
9. H. Rossi, et al., Health Phys., 8, 331 (1962).
10. K. Brien, et al., In book: "Neutron Dosimetry," 2, Vienna, IAEA (1963), p. 199.
11. J. Handloser, Health Phys., 2, 165 (1959).
12. M. Zel'chinskii, In the collection, 'Neutron Dosimetry," 2, Vienna, IAEA (1963), p. 397.
13. A. Sullivan and J. Baarli, Preprint CERN 63-17 (1963).
14. M. Zel'chinskii, V. N. Lebedev, and M. I. Salatskaya, "Pribory i teldmik eksperimenta," 6, 73 (1964).
15. M. Zel'chinskii, Nukleonika, 7, 175 (1962).
16. J. Baarli, Report CERN DI(HP) (1964).
17. J. Lehman and 0. Fekula, Nucleonics, No. 1, 35 (1964).
18. J. 'ruiner, et al., Health Phys., 10, 783 (1964).
19. P. P. Saksonov, et al:, Dokl. AN SSSR, 162, 688 (1965).
20. Yu. G. Grigor'ev, et al., In the book: "Biological Effects of Neutron and Proton Irradiations, 1, Vienna,
IAEA (1964), p. 223.
21. Collection: "Papers on the Biological Effects of High-Energy Protons" [in Russian], Izd. Instituta gigieny truda
i profzabolevanii Akademii meditsinskikh nauk SSSR, Moscow (1962), pp. 10, 65.
22, P. Bonet?Maury, et al., In book: "Biological Effects of Neutron and Proton Irradiation," 1, Vienna, IAEA
(1964), p. 261.
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23. J. Ashikawa, et al., In book: "Biological Effects of Neutron and Proton Irradiation," 1 Vienna, IAEA
(1964), p. 249.
24. V. S. Kiselev, et al., ZhE TF, 35, 812 (1958).
25. M. Zel'chinskii, Nukleonika, 10, 77 (1965).
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 list of the cover- to-
cover English translations appears at the back of this issue.
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HELICAL MAGNETIC CONFIGURATIONS WITH MINIMUM B
N. M. Zueva and L. S. Solov'ev UDC 533.9
A general consideration of magnetic configurations with helical symmetry and minimum l3-is pre-
sented. An approximate analytic expression is obtained for the specific volume V' (.I.) in the neigh-
borhood of the helical magnetic axis. Exact formulas for the specific volume V' ( 1.) and the mean
torsional angle of the lines of force i = 211X'(4') are given in terms Of single integrals. Graphs of
V' (t.) and x' (4') are plotted from numerical calculations of these integrals up to the separatix of
the magnetic surfaces.
An example of a straight periodic field with minimum -1-3, i.e., a field in which the specific volume V'(4')
d 1
on the axis of the system is maximum, was given in [1]. Furth [2], showed that a field with helical sym-
metry could also have a minimum B. The magnetic surfaces of such a field constitute a superposition of helical
magnetic tubes of constant cross section. We shall study helically-symmetric magnetic configurations in this paper.
General Considerations
An irrotational magnetic field B having helical symmetry may be described either by a scalar potential
B = Gl.max or a flux function ;G. In a cylindrical coordinate system r, z the helical field depends on two vari-
ables, B = B (r, 0), 0 = az, where a = 2r/ L, and L is the pitch of the helix. The general expression for the
scalar potential of a helical field has the form
CO
= Bzoz ? E zn (anr) sin ne,
?I=i
(1)
where zn are Bessel functions Zn(x) = anIn (x) + bnKn(x). Such a field constitutes a superposition of a homogeneous
field Bo parallel to the z axis, the field of a current-carrying filament extending along the z axis, and the field of
helical windings set on cylindrical surfaces r = Ri and r = Rg, i.e., we consider the Case in which R1 < r < R2.
The flux function is a combination of vector-potential components V) = A + ccrA9 and satisfies the equation
a t a2* 2(v2
Ta t 13 ar ao2 = 132
(2)
where a .1 + a2r2; in the case considered (irrotational field) I Bz + arB6 = const. The field components B are
expressed iii terins of zp:
r.13 7. = ;alp
cirl 3 ? .
ot
(3)
For a certain magnetic field B(r, 0) the function zp (r, 6) may be determined by integration with respect to r for a
fixed 0: = (arB z ? Bo) dr. The function 7,b corresponding to field (1) has the form
00
= Bzo ar 2
2 ? A 1n r ? (anr) cos nO.
n=
(4)
Translated from Atomnaya gnergiya, Vol. 20, No. 5, pp. 396-401, May, 1966. Original article submitted
August 14, 1965.
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Thus, we obtain exact expressions for the field and the magnetic
surfaces 0 = const in a cylindrical coordinate system. Let us derive
approximate analytical expressions for the specific volume and torsion
angle of the lines of force in the neighborhood of a helical magnetic
tube, and then exact integral expressions suitable for numerical cal-
culation of these quantities.
We note that the tangent of the angle between the helical line
and the z axis equals otr, and the field components (parallel to the
helical line 0 = const, and perpendicular to this and the radius vector
r) are respectively B11 = Ii3, B/ = /arra. The magnetic axis is the
helical line the pitch of which coincides with the pitch L of the field.
Fig. 1. Intersection of two neighboring Specific Volume of the Magnetic Tube
magnetic surfaces with the plane z =const.
Let the magnetic surfaces in the section z = const be closed lines
0(p, z) = const surrounding a point p = 0, displaced through a cer-
tain distance ro from the z axis. The longitudinal magnetic flux SC passing between two neighboring magnetic sur-
faces equals
6(1) = B1dS Bz6S,
(5)
where the integration is taken over the area SS of the section z = const (Fig. 1). The volume of the corresponding
layer is determined by integration over z:
6V = 6S dz = 6(1) dz
7131 ?
It follows that the specific volume V' ((I.) may be put in the form
dz
V' (CD) =
Bz
where
1
as BzdS
is the mean value of Bz over the area (SS.
(6)
(7)
(8)
A_,. Qd15
Since the element of area SS equals dS = p6 pd,9- = utp , then, if we take Bz = Bo + B1, where Bo =const
all)/aQ
is the axial field at the point p = 0, we get
27t
6S = 617 (115
diploe
0
2n 2n
Sip Q aft c4 08,
OS "z =_ OVIOQ 1j 7-- Os ,\ ova@
0 0
and for magnetic surfaces with helical symmetry the specific volume
V' (CD)
where L is the pitch of the helix reckoned along the z axis.
B,
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(9)
(10)
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For an approximate calculation of V' ( (13) let us put Bz and 0 in the neighborhood of the magnetic axis p = 0
in the form of series in g = p cos 0- and i p sin 0-, and let us confine ourselves to the ease in which the magnetic
surfaces are symmetric with respect to the E axis:
B -,-- Bo + b2i2 . . . ; (12)
(13)
+Ipire+Ap3v+v4w+ . ? .
in polar coordinates p, these expansions take the form
B = B0 + boQ cos 0 + (b1 cos2 b2 sin2 15) Q2
/30 -I- hi (15) +1105) --I- ? ? ? ;
11) = cos2 0 + p2 sin2 0) Q2+ (V3 tos30
cos* sin2 0) 03 . . . f (19) Q2
+ 12 (0)Q3 + ? ? ?
(14)
(15)
*112 f 211)
With an accuracy up to p3 we obtain Q1/ 2/1 . Hence, we find the following expression for Bz (valid
fi 2
in the neighborhood of the magnetic -axis):
where
271
( h2
= B 1/42
B o o ,s? 2fi
0
2n
SS 1 dO
Sal) 2 fi (0)
L .
The value of V() may be determined from the equation V'() + V" (cD0)? 0. Thus,
Bo
V"(cD)=
2n
L fSip h2 do.
Bg SS ) 2/1
0
On calculating the integrals in this expression we obtain
V" (W) ?i) os)2 4 (11Pct:205/2.
(16)
(17)
(18)
(19)
X [21i)z(b111'2? b211)1)? bo (31Pili)3 +11,44)1-
The location of the magnetic axis, or, in other words, the elliptic-type singular point of the family of lines 0 =const,
a a3b
z = const, is determined by the equations? = 0,-- = 0. The coefficients of the expansions (12), (13) are not alai-
vary. in bidet to determine the relations between these, we use expression (2). The coordinates g and n are re-
lated to r and 19 by the expressions (see Fig. 1)
=CP COSO r cost) ? ro; (20)
= Q sin* = r sin O. (21)
Expanding Bz and in powers of r-1.0 and 0 in the neighborhood of the magnetic axis 0' and using the equation
I 0'
Bz =? d E2 find a relation for the derivatives in the expansions for Bz and 0:
a + otr a anq. (), we on ? ?
446
B; ? " ? B" = ; E = ?
r z r .r z p
(22)
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v"
Fig. 2
Fig. 3
Fig. 2. Values of V", A, and B as functions of the parameter s, characterizing the eccen-
tricity of the normal cross sections of the magnetic surfaces in the case of an external cur-
rent winding.
Fig. 3. Cross sections of magnetic surfaces in the case of an externally-situated helical cur-
rent winding (40) with s = 0.7.
vI,, = -
4--2P p
r3 r2 ;
B0 ? (v" -173
B 11
-VIT ?
(23)
Here B11 is the value of the field at the magnetic axis, and the primes and dots indicate derivatives with re-
spect to r and 0, respectively, at the point 0', where 0' = = 0, r = ro. Thus, the expansions for Bz and 0 contain
altogether three free parameters; for these we may select, for example, , , . Substituting r = ro g --772/2r0,
0 R3 11/ro?Er1/r20 into these expressions apd omitting the index from ro for brevity, we obtain
11)
B,? Bo
a (10
2r r
= 1'1 V + 1P 112 V
2 2r2 6
)
12 + ;
tp" 4 ? 213
r3
f4')
(24)
(25)
By using these expressions we can write the formula for V" ( 4)) in terms of the indendent derivatives .0, 0", IP',
taken at the point P = 0:
r(q)) - 3 11) / 2 { 4a r2 31P-3
2B 0 (11)
r ) 2:1.13 p
iH ;2 ( t?
2 13r:1)
? ?
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(26)
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Fig. 4
Fig. 5
Fig. 4. Values of V' and x' as functions of the longitudinal
flux in the case of an external current winding (40).
Fig. 5. Disposition of closed magnetic surfaces with axis r =1
between helical current windings on cylinders r = R1 and r =R2.
In the neighborhood of the magnetic axis the helical tubes have elliptic sections; in the plane perpendicular
to the magnetic axis the ratio of the elliptical semiaxes equals
,b r2w,
a
2
The formula for V" ((L.) may conveniently be expressed in terms of a quantity s;
and it then takes the form
-=
b2 + a 2 ft
.1- -.?1P +frIP/r2
bi ?(11 v__137iivr2
2a VI ?82
V" (CD)
1 2 + 02
pt ?
X [ a2r2 (1 ? ' 1 + g) .
(27)
(28)
(29)
Since the first term in the square brackets is always positive (I e I < 1), then, in the case in which the normal cross
sections of the magnetic tube are circular (e = 0), the quantity V"( 4)) > 0. For an elliptical normal section (e 0)
the value of V" ( 43) may be negative.
Mean Torsion Angle i = ()
27r
For calculating x'(l,) we use the formula !I.?x -770 [3]relating the IPT1g4utlival f4Pc gid the azimuthal
flux x to the function 0 in the case of helical symmetry. Then we obton
x' (42)=-1? V' (a)),
If we limit consideration to the zero approximation only in the expansion of x' () in powers of then from
expressions (5), (9), we obtain
(30)
448
2n
61:130 B Q d* ?
VoV ? a laQ
(31)
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Fig. 6
Fig. 7
Fig. 6. Values of V", A, and B as functions of the longitudinal flux
internal current winding (41).
Fig. 7. Cross sections of magnetic surfaces in the case of an internally-situated current
winding (41).
e in the
case of an
then in analogy with the preceding case, we find
(cD) = 1? 171-82
?
(32)
Thus, in the neighborhood of the magnetic axis (for p -> 0) the mean torsion angle of the lines of force
i = 27r x' (fl is completely determined by the eccentricity of the elliptic cross section of the magnetic tube and the
inclination of the magnetic helical axis to the z axis of the cylinder on which the magnetic axis is wound.
Exact Expressions for V' (.1.) and ,i
In a case in which the helical magnetic field contains only the n-th harmonic, i.e., in a cylindrical system
of coordinates, the scalar potential has the form
Pm = Z + BZ,, (nr) sin nO (0 ?z), (33)
we can express the integral for V' (.t.) and x'(.1.) in explicit form. In expression (33) the longitudinal homogeneous'
field Bzo is taken as being equal to unity, and the unit of length is taken as 1/a = L/27r. The potential .A.(p corre-
sponds to the field of a current-carrying filament extending along the z axis: By, = ?Air. The last term in expres-
sion (33) describes an n-threaded helical field Zn(nr) = anin(nr) + bnKn(nr).
The flux function 0, corresponding to potential (33), has the form
r2
11) = + A In r ? BrZ;, (nr) cos nO.
(34)
If we use S(0) to .denote the area of the magnetic-surface cross section normal to the z axis, we can write
the following expressions for S'( 0) and (0):
r dr
S'(i)= avao ,
ar r dr (IP) = B, ova? .
The quantities V'(() and x' (43) are expressed in terms of S' (0) and I.' (0) by means of the formulas
V' (0) ?= 2rc (DS', ;
(35)
(36)
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2n
(37)
The axial field Bz = 1 ?nBZn(nr)cos ne. Let us introduce the notation F(r) = r2/2?A In r, f(r) = BrZ'n,(nr) and
express Op/t9 in the form of an explicit function of r:
F ? f cos nO, ?p/?8==nfsinnO
n
If we substitute these relations into expressions (35), we obtain the desired formulas for S. (0) and 4P(0):
rmax
2 r dr
S hh\
r min
rrnax
Zn (nr) (?F) dr
r Z;-1, (nr) V ? (1i) ? Fr
?
mm
(38)
(39)
In these expressions the integration is taken over r from rmin to rmax for which the expression under the root van-
ishes. On calculating these integrals, we obtain exact expressions for V1(4) and i = 2.7r x'( 49 as functions of 0.
Results of Numerical Calculations
By way of example, we considered magnetic surfaces formed by the fields:
Pm = Z ? B I (r) sin 0;
Z Acp ? BK (r) sin 0.
(40)
(41)
The helical field in the first example is created by a single-threaded current winding on a cylinder r = R2 (Figs. 2-4),
and in the second example on a cylinder r = R1 (Fig. 5). The radius of the center of the helical magnetic tube ro is
taken as unity, which corresponds to curvature k and torsion x, of the magnetic axis (k=a2r/13 =%, 1-t=c4/6 = %).
Figure 2 shows graphs of V"(), A, and B as functions of s. The values of V" were calculated from formulas
7213.Z? (tzar)
_= ,
(29), where The quantities A and B were determined from Eqs. (28) and conditions 0' (1, 0) =0.
arZn (nar)
The region of negative V" (4) corresponds to e > 0. The minimum V" corresponds to e = 0.7. For this case, we
constructed magnetic surfaces, the intersections of which with the plane z = 0 are shown in Fig. 3.
The exact values of V' ( 4) and x' (4.) calculated numerically from formulas (35) and (36) up to the separatrix
are shown in Fig. 4.
The values of V' and x' on the magnetic axis (r = 1) equal
and at the separatrix
2 1/-? ? '
V' (CD) =-n ; x, (43)) ? 1 Bo
(42)
V' (0)=24 , X' (0) =--- I, (43)
since 4' (0) -4. to, S' (0) to, and the value of V'('') is determined by the field Bs in an infinitely small region at
the edge of the separatrix (see Fig. 3, region around point rs).
450
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Fig. 8. Values of V' and x' as functions
?of in the case of an internal current
winding (41).
Figures 6-8 show the corresponding results for the second example
(41), the magnetic surfaces being constructed for the case & =-0.7.
In conclusion, we shall give some analytical expressions (general
for the case of helical symmetry) for the specific volumes on the mag-
netic axis and separatrix. Since the mean value of Bz on the separatrix
is determined by the region around the edges of the separatrix only,
we have
1 ' 1 -1 ?
B,= B,dS = ?n B Sh,
k=I
(44)
where the summation is taken over values of Bz on the edges of the sep-
atrix. Remembering the equality Bz = I/6 at the singular points (0' =0),
we obtain for V' (c1,) = L/Bz:
nL nL
---= ,
n 71
Xi I
L Bsit I y
k=1 = 6-Sk
k1
L LI30
V; -- ?
I ?
(45)
(46)
In the cases considered above, for which there is only one edge to the separatrix (n = 1), the ratio of the val-
ues of V' () on the separatrix and magnetic axis equals
v's 1+ a2r2s
Vj 1 ?a2r8 ?
(47)
From this we see that the magnetic configuration has a region of minimum B (at least near the separatrix), if
the distance from the axis of the helical windings to the edge of the separatrix rs is smaller than the distance to the
magnetic axis 1.0, i.e., in all cases in which the edge of the separatrix lies on the side of the axis of the helical mag-
netic configuration. There may be either a minimum or a maximum of B in the neighborhood of the magnetic axis.
Thus, our examination has shown that the possibility of the existence of magnetic configuration of the type
described, which a minimum of W in the neighborhood of the magnetic axis, is related to the spatial aspects of the
magnetic axis and the ellipticity of the normal cross sections of the magnetic surfaces (s 0).
LITERATURE CITED
. H. Furth and M. Rosenbluth, Phys. Fluids, 7, 764 (1964).
2. H. Furth, Lectures at the Seminar on Plasma Physics [Russian translation], Trieste (October, 1964).
3. A. I. Morozov and L. S. Solov'ev, Geometry of the Magnetic Field. In the book: "Questions of Plasma
Theory" [in Russian], No. 2, Moscow, Gosatomizdat (1963), p. 3.
451
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Declassified and Approved For Release 2013/04/01 : CIA-RDP10-02196R000700030001-4
PARAMAGNETIC EFFECT UNDER THE INFLUENCE
OF HIGH-FREQUENCY PRESSURE AND ELECTRON
PARAMAGNETIC RESONANCE IN PLASMA
V. M. Glagolev, I. N. Khromkov, UDC 533.9
and N. S. Cheverev
The interaction of uhf fields (co = 2.101? sec-1) in a space resonator containing dense plasma (n 1013
- 1014 cm-) in a steady magnetic field was studied experimentally. Under the influence of hf pressure
a paramagnetic current arises in the plasma; the associated effect of an increase in the static mag-
netic field inside the plasma agrees closely with the calculated relation.
For wHico = 0.5 paramagnetic resonance of the elections takes place; this leads to a sharp rise
87rpo
in plasma pressure Po' up to a = R., 0.2.
Ho
It is well-known that plasma situated in a trap formed by a magnetic field falling (on average) over the radius
is unstable with respect to flute perturbations. One means of stabilizing such convective instability is the pressure of
an hf field not penetrating into the plasma [1, 2]. Under the influence of hf pressure perpendicular to the direction
of the lines of force of the static magnetic field, a steady current develops on the surface of the plasma; this re-
moves the electric field of the flute perturbations and, thus, prevents the drift of plasma to the sides of the trap.
This surface current always acts in a direction such that the original magnetic field inside the plasma is increased,
i.e., there is a peculiar kind of paramagnetic effect.
Starting from the condition of equilibrium for plasma in hf and static magnetic fields (it is assumed that the
H Ho
electric hf field on the surface of the plasma is zero), for Po