SOVIET ATOMIC ENERGY VOL. 32, NO. 6

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• Publication Date: June 1, 1972
• Content Type: REPORT

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Russian Original Vo/.32, No. 6, June, 1972
Translation published December, 1972
SATEAZ 32(6) 533-630 (1972)
SOVIET
ATOMIC
ENERGY
ATOMHAH 3HEpria
(ATOMNAYA gNERGIYA)
TRANSLATED FROM RUSSIAN
rC
CONSULTANTS BUREAU, NEW YORK
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SOVIET
ATOMIC
ENERGY
.qoliiet 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 decreaselhe 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.
Editorial Board of Atomnaya Energiya:
Editor: M. D. Millionshchikov
Deputy Director
I. V. Kurohatov Institute of Atomic Energy
Academy of Sciences of the USSR
Moscow, USSR
Associate Editors: N. A. Kolokol'tsov
N. A. Vlasov
A. A. Bochvar
N. A. Dollezhar
V. S. Fursov
-I. N. Golovin
V. F. Kalinin
A. K. Krasin
A. I. Leipunskii
V. V. Matveev
M. G. Meshchery'akov
? P. N. Palei
V. B. Shevchenko
? D.L. Simonenko
V. I. Smirnov
,
A. P. VinogradoV"
A. P. Zefirov
Copyright 0 1972 Consultants Bureau, New York, a division of Plenum Publishing
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SOVIET ATOMIC ENERGY
A translation of Atomnaya Energiya
Translation published December, 1972
Volume 32, Number 6 June, 1972
Thermonuclear Wave of Combustion in a Limited Plasma -A.F.Nastoyashchii
and L.P.Shevchenko
Elementary Solution of the Neutron Transport Equation with Anisotropic Scattering
- N. V. Sultanov
On Estimating the Cost of Processing Liquid Radioactive Waste -B.S.Pavlov-Verevkin
and M.F.Krasnaya
Tissue Doses from High-Energy Nucleons - I.M.Dmitrievskii, Ya.I.Kabakov,
E. L. Potemkin, and V. V. Frolov
CONTENTS
Engl./Russ.
533 451
539 457
547,"463
549 465
ABSTRACTS
Computation of Thermal Constriction Resistance in the Presence of Contact of Laminas
Washed by Heat Transfer Agent - V.V.Kharitonov, L.S.Kokorev,
and V.K.Yurenko 554 471
Contact Conductivity of Cylindrical Bodies in Dense and Rarefied Gaseous Media
-N. N. Del' vin, L. S. Kokorev , Ya. A Bychko , and E .K. Shmachkov 555 471
A Method for Optimizing Certain Characteristics of Metal-Water Shields
- Yu. N. Borisov, A. A. Gordeev, and O. Ya. Shakh 556 472
Multigroup Heterogeneous Description of a Rod of Finite Radius - S.S.Gorodkov 557 473
Gasthermal Disintegration of Fuel Elements of a Water-Cooled Water-Moderated
Power Reactor with Separation of Fuel -A.T.Ageenkov and V.F.Savel'ev ? ? 558 474
Experimental Investigation of Contact Heat Exchange between Coaxial Cylindrical
Casings in Vacuum - I.I.Novikov, L.S.Kokorev, and N. N. DePvin 559 474
LETTERS TO THE EDITOR
Possibility of Neglecting Elastic Scattering in Calculating the Velocities of Nuclear
Reactions in a Uniform Sphere - V.K.Kapyshev and V.I.Sakharov 560 477
Fusibility Diagrams of Systems Involving Lithium Chloride, Uranium Trichloride,
and Uranium Tetrachloride - V.N.Desyatnik, N.N.Kurbatov, S.P.Raspopin,
and I.I.Trifonov 563 479
Optimization of a Reactor for Physical Investigations by Means of Fuel Profiling
- T. S. Zaritskaya, Yu. V. Petrov, A. P. Rudik, and E . G. Sakhnovskii 565 480 -
Use of Solid Track Detectors in Reactor Experiments - V.A.Kuznetsov,
A. I. Mogiltner, V. P. Koroleva, V. S. Samovarov, and L. A. Chernov 567 481
A New Intermetallic Compound in a Zirconium-Iron System - Yu.F.Babikova,
V.I.Filippov, and I.I.Shtant 570 484
Measurement of the Neutron Spectrum at the Outlet of the Horizontal Channels in a
Heavy Water Reactor and the SM-2 Reactor by the Time of Flight Method
- S. M. Kalebin, M. Adib, G. V. Rukolaine, 0. M. Gudkov, T. S. Belanova,
N. G. Kocherygin, and S. N. NikoPskii 573 485
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Study of the Dispersion (Particle-Size) Composition of Radioactive Aerosols in the
Layer of Air Close to the Earth -B.I.Ogorodnikov, 0.G.Skotnikova,
CONTENTS
(continued)
Engl./Russ.
V. I. Skitovich, L. S.-Soldaeva, and I. E Konstantinov
576
488
Interaction and Wettability in Uranium Dioxide and Zirconium-Based Melts
-A.T.Ageenkov, S.E.Bibikov, G. P.Novoselov, and V.F.Savel'ev
579
490
X-Ray Fluorescence Analysis of Heavy Elements by aX-Coincidence
- B. M.Aleksandrov, S. M. Solove ev, L. I. Tyvin, and V. P. hismont
581
491
Effective Neutron Absorption Cross Sections for Cf232 and Cf253 in the Central Channel
of an SM -2 Reactor - V A Anufr iev , V. D. Gavrilov, Yu. S. Zamyatnin,
V.V. Ivanenko, and G.N.Yakovlev ?
?
584
493
Unfolding Fast-Neutron Spectra from Threshold Detector Activities -A.M.Aglitskii,
S. S. Lomakin, A. G. Morozov, V. I. Petrov, and S. G. Popov
585
494
Angular Distributions of Photoneutrons from Al, Ti, Cu, Mo, W, and Pb
- V. P. Koval ev , V. P. Khar in, V. V. Gordeev, and V. I. Is aev
588
496
Calculation of the Boundary Effect in Particle-Transport Problems - S. I.Durinov,
V. V. Uchaikin, and A Koltchuzhkin
590
497
Methods for Calculating Nonstationary Processes Involving Separation of
Multicomponent Isotopic Mixtures in Packed Columns - I. A.Vereninov
and Yu. V.Rakitskii
593.--
499
Asymptotic Stability of a Reactor with Thermionic Converter - V.P.Gorbunov,
N. V. Isaev, O. V. Komarov, A. V. Kryanev, N. N. Petrov, A. F Piskunkov,
and S.B.Shikhov
596
501---
Calculation of Buildup Factors at Deep Penetrations -Sh.S.Nikolaishvili,
A I. Gabr ashvil i, N. N. D zhgarkav a , and E A Iordanishvili
600
504
Production and Acceleration of Multiply-Charged Phosphorus and Zinc Ions
- A.S. Pasyuk, Yu.P.Tret'yakov, and B.A. Zager
602
505
CHRONICLES
Meeting on Prognosis of Development of Nuclear Energetics - Yu.I.Koryakin
606
509
Exchange of Experience in the Field of Development and Utilization of Equipment of
a Water-Cooled Water-Moderated Reactor - A. P.Barchenkov
608
510
Diary of Collaboration
609
511
INFORMATION
The Thirty-First Session of the Scientific Council of the Joint Institute for Nuclear
Research (USSR) - V.A.Biryukov
611
513
Third All-Union Conference on Dissociating Gases - V.B.Nesterenko
616
516
Fourth International Conference on High Energy Physics and Structure of the Nucleus
- R. A. gramzhyan
618
517
International Conference on the Statistical Properties of Nuclei - U.G.Abov
and S. I. Drozdov
621
518
Portable Pulsed X-Ray Equipment "Kvant," IRA-3, RINA-1D - I. I.Komyak
and E.A?Peliks
623
520
On the Plan of Collaboration with the Atomic Energy Commission of the Iraqian
Republic - B. Yu.Golovanov
626
521
BOOK REVIEWS
N.M.Beskorovainyi, Yu. S.Belomyttsev, M.D.Abramovich, and V.K.Ivanov
-Nuclear and Thermophysical Properties, Fundamentals of Corrosion and
Fire Resistance [Vol. 1. Construction Materials of Nuclear Reactors]
-Reviewed by I. S. Lupakov
627
522
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CONTENTS
N.I.Chesnokov and A.A.Petrosov ?Systems of Exploitation of Uranium Ore Deposits
? Reviewed by I. F .Medvedev , N. S. Zontov, and L. Ch. Pukhall skit
The Russian press date (podpisano k pechati) of this issue was 5/25/1972.
Publication therefore did not occur prior to this date, but must be assumed
to have taken place reasonably soon thereafter.
(continued)
Engl./Russ.
629 523
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THERMONUCLEAR WAVE OF COMBUSTION IN A
LIMITED PLASMA
A. F. Nastoyashchii and L. P. Shevchenko UDC 621,039.616
Let us consider a thermonuclear plasma with a density close to that of solid matter. Such a plasma
may be created, for example, by heating a solid target with a laser pulse or electron beam. Let us suppose
that the plasma is surrounded by a material shell which prevents the escape of the plasma, and that energy
is released in it by virtue of a fusion reaction. At the same time, as a result of the contact between the
plasma and the walls restricting it, heat leaks away by thermal conduction. The radiation from the plasma
also causes a loss of heat.
Since the heat losses may be compensated by the thermal effect of the fusion reaction, it is reasonable
to consider the possibility of a self-sustaining reaction in which the "combustion" of the thermonuclear sub-
stance occurs in an almost steady-state manner. A problem of this kind was considered in [1] in relation
to three very simple plasma configurations: an infinite plane layer, an infinitely long cylinder, and a sphere.
It was shown that steady states could only occur in the presence of a specific relationship between the tem-
perature To in the center of the volume occupied by the plasma and the product of the density by the char-
acteristic dimension (nod).
There is furthermore a minimum possible value of the parameter (nod)cr for which thermal equilib-
rium may occur. For (nod) < (nod)cr a steady state of thermal equilibrium is impossible: the plasma initially
heated to thermonuclear temperatures will quickly cool. For (nod) > (nod)cr two states with different tem-
perature To may exist; however, only the state with the higher temperature is stable.
Presentation of the Problem
Let us consider a system similar to that just described. We shall assume, however, that only some
of the plasma is heated to the thermonuclear temperature, so that at the initial instant of time there is a
certain boundary between the hot and cold material (Fig. 1). The question arises: under what conditions (if
any) can this boundary move in the direction of the cold material, i.e., can a thermonuclear combustion
wave occur?
The heat-balance condition (heat-conduction equation) is [2]
?div q+Qs(p, 7'), (1)
where q is that heat-flow vector, related to the temperature gradient and the hydrodynamical velOcity of the
material v by
- )(AT+ pv (CpT +4) ; (2)
Qs is the amount of heat evolved by the heat source in unit volume in 1 sec; p is the mass of unit volume;
C is the specific heat at constant pressure.
Since a considerable part is played by the radiation loss QR in the plasma, the plasma being almost
transparent to this radiation for n 1022 cm-3 and T 108?K, we must include these losses in Qs(p, T):
Qs(P,T)=QP?Q11. (3)
In writing down the expression for the thermal effect of the reaction QF we must remember that, if
the dimensions of the plasma are small compared with the range of the charged particles constituting the
Translated from Atomnaya Energiya, Vol.32, No.6, pp.451-455, June, 1972. Original article sub-
mitted July 5, 1971.
o 1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York,
N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without
permission of the publisher. A copy of this article is available from the publisher for $15.00.
533
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w, cm/sec
2
108
8
6
107
A
Fig. 1 Fig. 2
Fig. 1. Layer of plasma contained by the walls AA' and BB'.
The region occupied by the hot plasma is shown shaded.
Fig. 2. Temperature 0 at the leading edge of the wave as a
function of the coordinate t
it 6 810
2 it 5 810
T
reaction products, then only a part of the energy evolved will
be used in heating the plasma [1].
The expressions for QF and QR may be found in the mono-
graph [3]. We shall assume that the thermal conductivity is a
power function of temperature
x=f3Th. (4)
For a Coulomb plasma in the absence of an external magnetic
field the power index in (4) equals k = 5/2, )6' = 2.1 ? 10-6 erg
?cm-1.sec-1.deg-7/2.
The divergence term in (2) describes the flow of heat
along the system and to the walls. The problem may be greatly
simplified if in order to describe the flow of heat to the walls
we make use of the expression derived in pi, which described
the flow of heat to the walls in an equilibrium state
2sI3 T k+1
(5)
Fig, 3. Dependence of the velocity of
the thermonuclear wave of combustion where s = 1 for a plane layer and s = 2 for an infinitely long
on To. cylinder. After introducing this simplifying assumption the
problem will essentially be one-dimensional. We shall seek a
solution to the heat conduction equation in the form of a plane wave moving at a constant velocity wl along
the x axis, i.e., in the form
T =T (x?w't), (6)
or in a system of coordinates moving with the wave in the form T = T(xl). As earlier, we denote the tem-
perature in the center of the plasma volume (this is obviously the maximum temperature in the system,
and thermal equilibrium exists at this point) by To and introduce the dimensionless temperature 0 = TT0-1
and the dimensionless coordinate t = xtd-1 (x ? wtt)d-1. We thus obtain a dimensionless heat-conduction
equation (here and subsequently ko are the power indices relating to the thermal conductivity along the sys-
tem; in general we may have k k11);
(6h1i00E + (0; nod, To)? (7)
Here
MC' QF-QR-Qx
v ? nocl?w'; s6=d2 ?
AToll 0411
no, is the density of the plasma immediately in front of the leading edge of the wave. In Eq. (7) we must re-
member that the density of the plasma is not constant, but temperature-dependent; it will later be shown
that the velocity of the wave is lower than the velocity of sound over a fairly wide temperature range, so
that the pressure may be regarded as constant, and then n 1/T.
Let us formulate the boundary conditions. We take the origin of coordinates in a system of reference
connected to the wave directly in front of the latter. Then according to Fig.2 the region with 0 > 0 occurs
in < 0. In the region t 0 we put o = 0. The second boundary condition for 0 = 0 corresponds to a zero
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-Y
a
Fig.4. Family of integral curves of
Eq. (22): a) v > Jim.; b) v < vet.; c) v
vcr?
energy flux. On the other hand, a fair distance behind the leading edge of the wave, in the region in which
thermal equilibrium is assumed to exist, the following two conditions must be satisfied: 0 = 1 and BE = 0.
These are readily seen to be equivalent. Thus in the region in which 0 P4 1 there is essentially one boundary
condition.
From Eq. (7) and the boundary conditions we may derive the condition for the existence of a thermonu-
clear wave of combustion, its velocity, and also the condition of thermal equilibrium behind the leading edge
of the wave.
Approximate Method of Solution
Thus the problem amounts to the solution of an ordinary differential equation (7) (nonlinear and non-
homogeneous). An analytical solution over the whole range of variation of the argument is hardly possible.
However, we may make use of an approximate method by seeking the solution in two limiting regions: 0 ? 1
and 9 1; the resultant solutions may be matched in the intermediate region at 0 rAl 1/2. At the matching
point two conditions clearly have to be satisfied; the two solutions must be equal and so must their deriVa-
tives (otherwise the condition of continuity imposed on the energy flux is infringed).
Solution of the Equation in the Region 0 ? 1. As already mentioned, immediately in front of the wave
we take 0 = 0. At the origin of the wave front the plasma clearly undergoes heating, mainly by virtue of
the heat arriving from the neighboring hotter region (intrinsic heat generation is as yet quite small here).
The heat arrives as a result of conduction; any heating due to the absorption of the intrinsic plasma radia-
tion may be neglected. The flow of heat not only heats the plasma but also compensates the losses due to
radiation and heat conduction to the walls and neighboring colder regions.
Neglecting the right-hand side of (7), describing the heat losses and intrinsic heat generation, by com-
parison with the first term on the left, describing the thermal flux from the hotter region, we have
(0k110t)t-I-vOt O.
?(8)
The solution of this equation satisfying the boundary conditions is
1/k11,
(9)
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The minus sign occurs because owing to the coordinate system chosen the region 9 > 0 corresponds to nega-
tive so that the quantity in the brackets is positive. Substituting the solution into Eq. (7) we readily see
that in the absence of marked radiation losses the right-hand side of (7) may indeed be neglected. In the
presence of such losses we may only neglect this for k11 > 5/2. In the case k11 = 5/2 the solution has the
same power dependence on t as that given by (9) if in the latter we take k11 = 5/2:
0
but with a different factor in front of one allowing for the radiant energy losses
q+=-1v+ 17.2f65(v)2+--25 OR (1).
(10)
We see from the resultant solutions that the wave front is curved (Fig. 2) on approaching the point t
= 0: Ot I t
Solution of the Equation in the Region 0 ki 1. In this case we may certainly not neglect any term on
the right of Eq. (7); however, the equation may be greatly simplified by introducing a new function z in ac-
cordance with
z=1-0 0, and this can only occur if
(17)
From physical considerations it is easy to see that for 44(1) > 0 there can be neither stable thermal
equilibrium behind the leading edge of the wave [1] nor even a combustion wave at all. If 0(1) < 0 there
can only be a cooling wave. We may convince ourselves of this by considering that the cooling wave corre-
sponds to v < 0 (it must clearly travel in a direction opposite to the wave of combustion); then p+ > 0. An
additional condition for the existence of a cooling wave, as indicated by (10) and (11), is the existence of
radiation losses.
Finally for the region 9,-":4 1 we obtain the solution
0=1? C+04i. (18)
The constant in (18) cannot be determined from the boundary conditions, since in the region 9RI 1 there is
essentially only one boundary condition, which has already been used in determining C_.
Matching of the Solutions and Velocity of the Wave. As already mentioned at the beginning of the sec-
tion, the solutions have to be matched at 9 1/2, i.e., where the intrinsic heat generation is small, but not
so much so as to be negligible by comparison with the flow of heat from the hotter region. At the matching
point two conditions arising from the continuity of the temperature and its derivative have to be satisfied.
From the first condition we find the integration constant C+ and from the second the velocity of the wave.
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The simplest expression for the velocity of the wave is obtained when the radiation losses are insignificant
(in the region T > 108?K in which a combustion wave is possible, this is in fact so). If k 5/2 we have for
the wave velocity
PTIOII VI 45' WI
mCp nod
(19)
Here w is the velocity of the leading edge of the wave with respect to the hot material, i.e., w = (n/n0);
the numerical factor in general depends on the power index kii. Comparison with the exact solution ob-
tained numerically shows that (19) correctly describes the change in wave velocity if a = 1.
If we then use the relation between nod and To imposed by the condition of thermal equilibrium behind
the leading edge of the wave, we may find the velocity of the wave as a function of To. For the case in which
thermal contact of the plasma with the walls is effected by the ions (a fairly strong magnetic field being ap-
plied parallel to the walls) while heat is transferred along the system by the electrons in accordance with
Eq. (4), in which kl1 = 5/2, this relationship is illustrated in Fig.3. The velocity of the wave rises quite
quickly with increasing temperature, but still remains below the velocity of sound up to temperatures of
,-4.108?K.
Study of the Problem by the Phase-Plane Method
The phase-plane method [4] easily enables us to draw conclusions regarding the possible existence
of a thermonuclear combustion wave and also to make a quantitative estimate of the wave velocity. Further-
more this method is useful in that it clearly demonstrates the uniqueness of the resultant solution.
Introducing the phase variable y(0)= 8, we transform the original Eq. (7) to the form
y2 + v CA (0 )
(20)
kIIehlr kHell? 1 kII ?"?'
where 0(0) is the thermal source function including the generation of heat, and also the loss of heat at the
walls and by radiation. If we then use the functions
y (0) _ V ?1? + ilek11-1 ) Vi V \ 2 gb (0)
(21)
kII0111 k kII
that is, the roots of Eq. (20) without the right-hand side, the latter may be transformed into
k ,
Ye = ? *[Y?Y+ [y y- (0)],
(22)
The family of integral curves relating to Eq. (22) for k11 = 5/2 is shown schematically in Fig. 4. The
thick lines represent the functions y = y?(0). It is quite easy to show from the results of the previous sec-
tion that the desired intergral curve for 0 ?1 is linearly related to y_(61) thus: y(0) e?-zi cy_(0); the proportion-
ality factor c > 1. For 0 1 also we have a linear relation y = cy,(0), but here c < 1. Thus the integral
curve representing the solution of the problem should proceed below the special curve y_(0) for 0 ?1 and
above y?(0) for 0 :--t$ 1. The course of the integral curves varies considerably with the relation between v
and ver where yer is the value of v, for which the expression under the root in (21) is not negative for any
from the range [0, 1].
Figure 4a corresponds to v > ver. The integral curve satisfying the boundary condition 9=1, 9= 0
for t = -co should arise from the point at which 0 = 1; y = 0 (point B). We see from Fig. 4a that the required
integral curves connecting the "cold" (0-.- 0, 0) and "hot" states of the material are nonexistent in this
particular case.
Figure 4b shows the integral curves for v < vcr. We see from this figure that in this case also there
are no integral curves satisfying the conditions of the problem.
The integral curve (BC) does, however, exist for v fkqvcr (Fig.4c). From this we find the velocity of the
wave, which agrees qualitatively with the value obtained by the approximate method in the previous section.
The necessary condition for the existence of a thermal combustion wave is also confirmed, namely, that the
derivative of the right-hand side of Eq. (7) should be negative at temperatures close to the maximum tem-
perature in the plasma.
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We may thus conclude from the foregoing analysis that a combustion wave can only occur in a thermo-
nuclear plasma for plasma parameters which make the temperature derivative of the effective heat source
(including the energy effect of the fusion reaction and the heat losses associated with radiation and transfer
to the walls) negative. This criterion is simultaneously the condition for the stability of the thermal equilib-
rium in plasma between walls [1]. The foregoing solution with the monotonic wave profile (Fig. 2) is the
only one; there are no other types of wave in the present case.
Under different conditions waves of other structures may possibly arise. Thus it was shown in [5] by
a numerical technique that in an unbounded plasma in which the plasma had no thermal contact with any
walls a combustion wave of the space-periodic type was quite possible.
LITERATURE CITED
1. A.F.Nastoyashchii, At. Anerg., 32, 43 (1972).
2. L. D.La.ndau and E.M.Lifshits, Mechanics of Continuous Media [in Russian], Gostekhteorizdat, Mos-
cow (1945).
3. L.A.Artsimovich, Controlled Thermonuclear Reactions [in Russian], Fizmatgiz, Moscow (1961).
4. N. N. Bogolyubov and Yu. A. Mitropollskii, Asymptotic Methods in the Theory of Nonlinear Oscillations
[in Russian], Fizmatgiz, Moscow (1958).
5. S.Alikhanov and I.Konkashbaev, Nucl. Fusion, 11, 119 (1971).
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Declassified and Approved For Release 2013/03/01 : CIA-RDP10-02196R000300100005-6
ELEMENTARY SOLUTION OF THE NEUTRON TRANSPORT
EQUATION WITH ANISOTROPIC SCATTERING
N. V. Sultanov UDC 539.125.52
The usefulness of a system of elementary solutions of Boltzmann's equation when considering neutron-
transport problems was demonstrated in [1]. Assuming plane geometry and isotropic scattering, a com-
plete system of elementary solutions of the one-velocity steady-state Boltzmann equation was there derived,
the distribution function 11/(x, it) depending on two variables. The case of anisotropic scattering was studied
in [2, 3].
In certain problems of practical interest one has to solve the Boltzmann equation for a distribution
function depending on three variables (for example, in considering the oblique incidence of a parallel neutron
beam on a plate, or in setting up a system of elementary equations possessing cylindrical symmetry from a
system of elementary plane solutions [41). In this case new solutions appear, allowing for azimuthal devia-
tions from symmetry. The case of isotropic scattering was considered in [4].
We shall now study the case of anisotropic scattering. The scattering indicatrix is expressed in the
form of a finite series in Legendre polynomials;
1
(120')?
k=0
(1)
Whereas in the isotropic approximation the solutions responsible for the azimuthal asymmetry have only a
continuous spectrum of v values, in the anisotropic case the new solutions with numbers m s N acquire a
discrete spectrum of vr. In order to find the vr transcendental characteristic equations will be obtained
in this paper. The number of yr will be estimated. We shall also demonstrate the orthogonality and com-
pleteness of the resultant system of elementary solutions.
The system of plane elementary solutions thus obtained is suitable for modifying the computer pro-
gram "Praktinets" also used for calculating the cell characteristics in the linear-anisotropic approxima-
tion. We shall calculate several cells by means of this modified program.
System of Plane Elementary Solutions. The one-velocity Boltzmann equation for the distribution func-
tion ?1,(x, co) in plane geometry takes the form
.9T , +w (x, p,, (p) = C f (5252') W (x, 52') do', (2)
where c is the number of secondary neutrons per collision; x is the distance measured in neutron free paths.
We shall seek a solution in the form of a plane wave;
(x, t,(p) = e-xivcD (v, t, (p), (3)
where v is a complex number. Substituting (3) into (2) we obtain an equation for 4)(v, jt,(p):
(I - 14) (v, )=c52 f (MY) (I) (v, 52')d2'. (4)
We use proposition (1) and the addition theorem for Legendre polynomials twice [5]. Then (4) takes
the form
Translated from Atomnaya Energiya, Vol.32, No.6, pp.457-462, June, 1972. Original article sub-
mitted July 19, 1971.
1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York,
N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without
permission of the publisher. A copy of this article is available from the publisher for $15.00.
539
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FA
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TABLE 1. Results of the Calculation of Some Cell Characteristics
Number of cell
Characteristic
Approximation
isotropic
linear-anisotropic
GI q3
. transport
Gt G3
Gt
G3
1
52/51
1,550
1,548
1,488
1,487
1,511
1,513
6
0,9189
0,9190
0,9219
0,9220
0,9208
0,9207
9
1,296
1,273
1,296
1,273
1,296
1,281
a5s/C
1,713
1,661
1,651
1,599
1,674
1,630
6
0,8749
0,8779
0,8778
0,8808
.0,8767
0,8791
3
(132gfit
1,598
1,581
1,501
1,484
1,509
1,496
(DOD'
2,322
2,297
2,127
2,102
2,142
2,113
e
0,7964
0,7983
0,8092
0,8110
0,8082
0,8101
Note: cbi is the mean flux in the i-th zone; % is the number of the approximation [7].
1 2n N N
'In.(4)-(V) -imr(T-cr)
(1?E) CD (VI 11, (1)) = c S 2 [ 2 b. ) P (1.0 e (v, (r)dd.
lil(p'
4a (5)
-1 0 m'=.0 n=m'
Here ?, 12' are the projections of SI and al respectively on the x axis; co, co' are the azimuthal angles of St,
Sr in a plane perpendicular to the x axis.
Let us expand N
(v, 1.1.) = 6 (v? ?).
The complete solution of the Boltzmann equation (2) takes the form
where
00 x
11r(x, (p)? E s am (v) CD' (v,
m=?oo V
mut
f (v) dv.
.?mm
(31)
(32)
(33)
The coefficients am(v) are determined from the boundary conditions.
Number of Discrete Elementary Functions. The function Sen(z) is an analytical function in the ex-
panded complex plane with a cut along a section of the real axis [-1, 1] without any singularities. Hence
the number of roots of this function may be determined by means of the principal of the argument
2Mm =-- arg (z),
(34)
where the contour c passes around the section [-1, 1].
Carrying out operations analogous to those of [3], for the case in which S2In+(v) and S2m-(v) are not
equal to zero in the section [-1, 1], i.e.,
Stm (v) = [Pam (v)]2+ [ n?v Nm (v)]2 0 (35)
(36)
we obtain for v E [-1, 1],
Here the signs ? relate to values of the function above and below the cut respectively.
Let us estimate the number of discrete values of yin. Since Nm(v) is an even polynomial of order 2N,
the number of roots of the polynomial vNm(v) in the section [0, 1] cannot exceed N + 1; there is no less than
one root for v = 0 and no less than m roots for v = 1 (see Eq. (26)).
Thus
m+14Mm O.
By making a change of variables [1]
x = y = r 171 ? p,2
we reduce this to an ordinary differential equation
01Y+
T+ ( 1/R2? y2, y) = 0; 0 < y < R.
The solution of this equation takes the form
?rIA+R 17 1? ( ;-02
T+ (r, 1.0= Et
exp (? Eap) dp
0
=-21.-2 11? exp [ ?Ea ? rp, R17 1? (Ter )2(1 ? 0)) / ?
Using Eqs. (5) and (6) we find the error 6Q1 in the calculation of the functional (1):
+1
OQi f.). 16L (To)1I (r, ?)] dV d52? S exP ( Ear) S [6 (1?)?g (?)]
r2
X 11 ?exp [_Ea ? rp, R 1? (4 ) 2 (1 2)) dp,r2 dr.
(6)
(7)
We note that the resultant equation (7) may be used if the perturbation (3) is small. This occurs if, for
example, the elastic scattering of the neutrons takes place (as a result of anisotropy) close to the direction of their
initial approach. Then by expanding the second factor in the integrand in series in powers of (1 ? ?) we obtain
? exp [_Ea ( R 17 1? ( Ter ) 2 (1? t/2))= exp ( (R?r)) {1 ? Zar (1 ? Tir ) (1-0
?Za [ra ( 1 ? + R ( )2 _1(1-21'2)
Confining attention to the first two /terms and substituting the resultant expression in (7), after integra-
tion we obtain
+1
)?Z45(2i EiEs [-6- exp (? EaR)] [1? lug (?) dd.
(8)
As indicated by physical considerations, the error in fact tends to zero when R tends either to zero or
to infinity. The maximum error occurs for R = 2/Ea.
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By way of example let us consider a lead sphere with 14.06 MeV neutrons coming from a monoenerge-
tic point source. Then according to [3]
1? pg (1) d=0.1; IT? Ea =0,36.
?1
In this case the error in determining the rate of absorption of the neutrons in the whole sphere reaches its
maximum for R = 11.78 and equals oQi = 0.006. If in order to calculate the functional (1) we use Eq. (8),
then Qa = 0.865, i.e., thc error in determining Qa is 0.7%.
The permissibility of neglecting elastic scattering is confirmed experimentally. Using copper in-
dicators with the reaction Cu63(n, 2n)Cu62, the distribution of neutrons from a point neutron source of energy
14.1 MeV in an infinite boron carbide sphere was studied in [4]. Since in this case the anisotropy of elastic
scattering was also large, while the cross section of the reaction Cu63(n, 2n)Cu62 for energies below 14.1
MeV was small, on neglecting the elastic scattering the neutron flux of this energy should be given by Eq.
(4). In Fig.1 the continuous curve gives the flux p1(r) measured with a copper detector in relative units:
(pi (pi (r) 47tr2
' (pi (ri)4nri '
where ri = 6 cm. The broken curve represents the function
+1 exp HET xs) rl (I dp,
4nr2 4nr2
?1
exp [? (ZT Es) ril
The calculated curve may be considered close to the experimental curve on allowing for the accuracy of the
experiments as well as the accuracy of determining the inelastic scattering cross section. The course of
the experimental curve (exponential fall) confirms the validity of the neglect of elastic scattering.
LITERATURE CITED
1. G.I.Marchuk and V. V.Orlov, in: Neutron Physics [in Russian], Gosatomizdat, Moscow (1961), p.31.
2. G. I. Marchuk, Numerical Methods of Calculating the Energy of Reactors [in Russian], Atomizdat, Mos-
cow (1958).
3. I. V.Gordeev, D. A.Kardashev, and A. V.Malyshev, Nuclear-Physical Constants [in Russian], Gosato-
mizdat, Moscow (1963).
4. S.P.Belov et al., At.tnerg., 6, 663 (1959).
562
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Declassified and Approved For Release 2013/03/01 : CIA-RDP10-02196R000300100005-6
FUSIBILITY DIAGRAMS OF SYSTEMS INVOLVING
LITHIUM CHLORIDE, URANIUM TRICHLORIDE,
AND URANIUM TETRACHLORIDE
V. N. Desyatnik, N. N. Kurbatov, UDC 546.791.6:131.34
S. P. Raspopin, and I. I. Trifonov
The use of melted salts as electrolytes in obtaining metals by electrolysis makes it necessary to study
the fusibility diagrams of binary and ternary systems. In the present article we shall discuss the results
of investigations conducted on binary and ternary systems involving uranium trichloride, uranium tetra-
chloride, and lithium chloride.
The binary systems LiCl?UCIs and LiCl?UC14 have been studied previously [1]. However, a check
revealed some discrepancies between the published data and the results of the investigation. The phase
diagram of the binary system UC13?UC14 was published in [2].
In order to conduct the experiment, we prepared pure initial substances [3, 4]. The uranium tetra-
chloride had an atomic ratio of [Cl] [U] = 4.01 and a melting point of 590 ? 2?C. The uranium trichloride
.00 s
00.44. SON*
A.* cr:?
LiCL%
*4,
I.
4r ? .1C4i
rtr
? P*4P 4%* ,
f4it, ? \ ?
C13
0 4
irsownevis
.
. . . ? am
.t....ezurawsionaum
111111511M
1,....
z
8
ME
00
00 n
0 E.
0 cc,
0
Fig.l. Fusibility diagram of the system LiCl?UC13?UC14 and the cor-
responding phase diagrams of the binary systems.
Translated from Atomnaya E.nergiya, Vol.32, No.6, pp.479-480, June, 1972. Original article sub-
mitted June 9, 1971.
o 1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York,
N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without
permission of the publisher. A copy of this article is available from the publisher for $15.00.
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Declassified and Approved For Release 2013/03/01 : CIA-RDP10-02196R000300100005-6
prepared from the tetrachloride thus obtained [3] had an atomic ratio of [Cl]: [U] = 3.02 and a melting point
of 835? 2?C. Anhydrous LiC1 was obtained by dehydrating OSCh brand lithium chloride in a vacuum at
temperatures of 50, 100, and 300?C, maintaining it at each temperature for 4-5 h. After this the lithium
chloride was heated to 750?C, and gaseous chlorine obtained by the electrolysis of lead chloride was bubbled
through the melt for 30-60 min, After the chlorine had been bubbled through, the melt was slowly cooled
for 2 h in a vacuum. The resulting LiC1 was analyzed for chlorine and lithium content; these were found to
be stoichiometric within the limits of error, The melting point of this LiC1 was 605 ? 2?C, which is in good
agreement with the published value [5].
The starting materials were kept in a dry chamber, since they are extremely hygroscopic.
The binary and ternary systems were studied by the method of differential thermal analysis, with the
cooling curves recorded at a rate of 5-8 deg/min on an NTR-62M recorder.
The binary system LiC1?UCI3 plotted on the basis of our investigations is characterized by one eutec-
tic with a melting point of 490 ? 2?C at a UC13 content of 26.5 mole % (see Fig. 1), which is in good agree-
ment with the results of [1]. In the binary system LiCI?UC14 we found two eutectics at UC14 content values
of 31.00 and 44.00 mole %, having melting points of 415 ? 2 and 400 ? 2?C, respectively, and the compound
2LiCI.UCIA, which melted congruently at 430 ? 2?C. Unlike [1], which does not contain any investigation of
the UC14 = 75-100 mole % region, our study revealed the existence of a UC14?UC13 solid solution. In the
ternary system LiCI?UC13?UC14 (see Fig.1) we studied 10 temperature sections passing through the vertex
of the concentration triangle (corresponding to the composition LiC1) and the opposite side. In this system
we found four crystallization fields: UCI4, 2LiC1?UC14, LiC1, and UC13. The quasibinary cut UC13-2LiC1
? UC14 divides the system into two secondary simple eutectic systems. The eutectic points (E4) formed by
the crystallization fields UC14, 2LiC1 ? UC14, and UC13 is characterized by a melting point of 290 ? 2?C and
corresponds to a composition of 50.00 mole % LiC1, 10.00 mole % UC13, and 40.00 mole % UC14. The crystal-
lization fields 2LiC1?UC14, LiC1, and UC13 converge, forming a eutectic (E2) with a melting point of 350 ? 2?C,
corresponding to a composition of 68.00 mole % LiC1, 13.00 mole % UC13, and 19.00 mole % UC14.
LITERATURE CITED
1. Reactor Handbook, Vol.1, Materials, New York (1960), p.425.
2. V. N. Desyatnik et al., Atomnaya Energiya, 26, 549 (1969).
3. J.J.Katz and E.Rabinovitch, The Chemistry of Uranium [Russian translation], Izd-vo Inostr, Lit.,
Moscow (1954).
4. V.N.Desyatnik, I.F.Nichkov, and S.P.Raspopin, Izv.Vuzov.Tsvetnaya Metallurgiya, No.5, 95 (1969).
5. Handbook of Melted Salts [Russian translation], A. G.Morachevskii (editor), Khimiya, Moscow (1971).
564
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Declassified and Approved For Release 2013/03/01 : CIA-RDP10-02196R000300100005-6
OPTIMIZATION OF A REACTOR FOR PHYSICAL
INVESTIGATIONS BY MEANS OF FUEL PROFILING
T. S. Zaritskaya, Yu. V. Petrov, UDC 621. 039.51
A. P. Rudik, and E. G. Sakhnovskii
In [1] a functional of the following form was proposed for estimating the reactor component of the cost
of obtaining information in experiments in a stationary reactor:
P jI _L \
(1)
IED (rMaX) yG P 1 '
where P is the power of the reactor; y is the average burnup of the fuel elements at the time of their re-
moval; (D(r) is the flux of thermal neutrons; rmax is the maximum point of the function c(r) in the reflector;
G is a given constant. It would be of interest to find a distribution of nuclear fuel in the volume of the reac-
tor active zone such that the functional (1) will be a minimum.
For a qualitative investigation of the solution, let us consider the simplest model of this problem.
We shall confine our attention to the case in which the nonuniformity of the burnup may be neglected. We
select a plane symmetric reactor with an active zone of halfwidth R and a reflector of thickness A. We as-
sume that the structural materials (including the moderator) are uniformly distributed through the volume
of the reactor and that their properties are constant both in the active zone (0 r R) and in the reflector
(R r R + A). If we assume that resonance absorption and neutron capture in the slowing-down process
are insignificant, the system of equations for the thermal-neutron flux (NO and the fast-neutron flux co(r) in
the active zone can be written in a two-group diffusion approximation as follows:
1+ u D II 1 riu
?VP-1-13TS(D=0,
(2)
where D = Df/Dt is the ratio of the diffusion coefficient of the fast group to that of the thermal group; Lo is
the diffusion length when absorption in the uranium is disregarded; T and ti are, respectively, the square of
the slowing-down length and the effective number of generated fast neutrons per capture in the uranium.
The equations for the neutron fluxes in the reflector can be obtained from the system (2) if in that system
we take u 0 and replace Lo, T, and D by the reflector constants L*, T*, and D.
The function u(r) is the ratio of the macroscopic cross sections of absorption of the uranium and the
structural materials and is proportional to the uranium concentration, which is not known in advance. How-
ever, it is subject to the following technological and heat-engineering limitations:
o< u (r) Pi and u(r) = Umax we would have at the boundary with the reflector a segment with (D(r) > qmax
/Umax, on which the heat-engineering limitation would be violated. Therefore if P > P1, the optimal dis-
tribution must be one with u(r) =Umax at the center of the reactor and u(r) = qmax/cb(r) at the boundary
with the reflector. As the power increases further, at P = P2 > Pi we will find 43(0) = qmax/Urnax, and for
Pi > P2 the optimal combination would be one with segments u(r) = qmax/t.(r) at the center and at the edge
and with a segment u(r) =Umax between them. Lastly, at a power-value of P = Ps ,> P2 the segment with
u(r) = Umax will shrink to a point, and when P 133, the optimal combination will be one with u(r) = qmax
/c1)(r) everywhere in the active zone,
Numerical calculationst confirmed that in the cases indicated the necessary condition for optimality
of the system under consideration is satisfied. The calculations were carried out for n = 2,3, Lo = 169 cm2,
L*2 = 104 cm2, = 144 cm2, A = 200 cm, and equal diffusion coefficients in the active zone and in the
reflector; (I)(r) was normalized so as to make qmax = 1.
Table 1 shows for purposes of illustration the variation of 0,5
7,8
2,0
1,5
0,5
14,7
J
*Particle density taken as 2 gicrns.
,.+.-----T
+To
v 0
2
1
g
n
0/
0 01 0,2 03 0,4 0,5 r; p
Fig.2. Dependence of the aerosol trapping
efficiency on particle radius (v in m/sec):
1) FPA-70-0.12, v = 1.0; 2) FPP-70-0.25,
v = 0.75; 3) FPP-70-0.25, v = 1.0; 4) FPP-
70-0.25, v = 1.8; 5) LFS-2, v = 1.0.
resolution
Ci.
of the spectrometer was 1%, the
determined nephelometrically. Figure 1 shows the depen-
dence of the filtering coefficient a = -logIc/pp] (here [isp]
is the resistance of the filter for a velocity of 1 cm/sec)
on the rate of filtration for FPP-70-0.25. Analogous re-
sults were obtained for FPA-70-0.12.
For velocities up to 40-60 cm/sec the aerosols are
mainly trapped as a result of the diffusive settling of the
particles and direct capture. With increasing velocity in-
ertial settling begins, and above 100 cm/sec a becomes
proportional to the square of the particle radius. In the
filtering velocity range studied LFS-2 trapped the graded
aerosols almost completely (Fig.2).
Samples of air close to the ground (60-70 thousand m3)
were taken in Podmoskov in April-May,1970 on days in
which there was no fallout. The area of the filter was 0.5
m2 and the rate of filtration 0.75-1.8 m/sec. The number
of layers was 3-4. After air had ceased passing through,
each layer of the filter was compressed into tablets
cm in diameter and measured in a semiconductor 'y-spec-
trometer with a Ge -Li detector 20 cm3 in volume. The
measuring time for each layer 5 h, the activity 10-3-10-10
The dimensions of the particles and the mean efficiency of particle trapping were determined from
Fig.2. The results were analyzed by a matrix method. Let us use qi to denote the activity of the particles
in the j-th size range and (pi to denote the ratio of the activity of the i-th layer in the filter to the total ac-
tivity of the sample; nii denotes the average efficiency of the i-th layer of the filter for the j-th range of
size. Then for a three-layer filter we may write the following system of equations
{(pi=giiqi-1-812q2+ 813q3;
(P2- (1-- 811) 821q1?(1- 812) 822q2+ (1- 80 823q3;
(P3- (I - 811) (1- 820 E3iqi + (1- 810 (1- 822) 832'72+(1 - 813) (1-823) 833q3.
The solution of the system of equations for qi, q2, and q3 for each isotope gives the proportion of the
total activity of the isotope concentrated in the particles of the size range selected, as in [21 (Table 1).
It follows from the results that, at the time of sampling the air close to the ground, most of the radio-
active isotopes in the global fallout lay in particles with radii of under 0.5 ?. The isotopes Cs137, Rum, and
Rh106, having volatile ancestors, and also the cosmogenic isotope Be7, were concentrated in far smaller par-
ticles than Ce144, Zr35, and NO. The agreement between the particle distribution of Zr35 and its daughter
isotope N1335 indicates that the method of determining the dispersion composition of the radioactive aerosols
by filtration in multilayered filters was fairly accurate. The relative means square error for all the iso-
topes studied, obtained by averaging the results over three or four samples, was 15%, except for the frac-
tion with r > 0.5 11, for which only rough values were obtained (error -100%).
The results correspond to existing data regarding the formation of radioactive aerosols in the cloud
of a nuclear explosion. According to these views, the isotopes having volatile ancestors should concentrate
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in the finer particles.- After radioactive aerosols have passed into the atmosphere, the differences in the
sizes of the particles (independently of the isotopes concentrated within them) arise mainly from processes
of washing out and coagulation with natural aerosols [3, 4].
Residual differences may be explained in two ways; first, the samples were taken at the spring maxi-
mum, when intensive exchange with the stratosphere was taking place; secondly, at this period "fresh" fis-
sion products from the nuclear explosion of September 29, 1969 were falling out [5].
The authors wish to thank G.V.Sukhov for help in the measurements with the semiconducting spectro-
meter.
LITERATURE CITED
1. I.V.Petryanov et al., Fibrous Filter Materials FP [in Russian], Znanie, Moscow (1968).
2. L. Lockhart et al., J.Geophys.Res., 70, 6033 (1965).
3. B.Shleen et al., Science, 147, 290 (1965).
4. Ya.I.Gaziev, in; Radioactive Isotopes in the Atmosphere and Their Use in Metrology [in Russian
Atomizdat, Moscow (1965), p.153.
5. R.Weinrecih et al., Z.Naturforsch., 25a, 1156 (1970).
578
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Declassified and Approved For Release 2013/03/01 : CIA-RDP10-02196R000300100005-6
INTERACTION AND WETTABILITY IN URANIUM DIOXIDE
AND ZIRCONIUM-BASED MELTS
A. T. Ageenkov, S. E. Bibikov, UDC 621.039.54
G. P. Novoserbv, and V. F. Savel'ev
Thermal bi.eaching of fuel elements using oxide fuel through the melting of their steel jackets was
discussed in [1]; it would be desirable to use this method on fuel elements with zirconium jackets as well.
However, attempts to melt zirconium jackets directly have not yet brought the desired results because at
and above the melting point of zirconium (1860?C) there is intensive interaction between the uranium dioxide
and the melt. At the same time, eutectic alloys of. zirconium with such metals as Be, Fe., Ni, Cu, Cr, and
others have much lower melting points (930-1300?C) [2].
Investigations have therefore been conducted on the conditions of the alloying of zirconium with various
metals, as well as on the wetting and interaction taking place in a system consisting of a metal alloy, and
solid uranium dioxide. The added material must satisfy the following conditions: alloyability with the zir-
conium jacket at a relatively low temperature; oxygen affinity lower than that of zirconium; low volatility;
easy availability; and relatively low price. On the basis
of the above criteria, we selected copper, iron, nickel, and
stainless steel for our investigation. Iron forms two eutec-
tics with zirconium [at 934?C (16% Fe) and at 1330?C (84%
Fe)] and an intermetallic compound Fe2Zr (1600?C) [2].
Zirconium alloys cOntaining more than 15% copper, are in
the liquid state at temperatures above 1300?C [3]. Nickel
and zirconium form two eutectics rat 960?C (15% Ni) and
1340?C (84% Ni)] and an intermetallic compound Ni3Zr
(1700?C) [4].
In our experiments we investigated zirconium iodide,
an alloy of zirconium with 1% niobium, iron (St.3 steel),
TABLE 1. Amount of Uranium Going into
the Melt and Contact Angles of Wetting
(Exposure time 30 min)
.1 Z
ij c.
OA
?6
Additive
? dditivem
the .
.1loy ?
elt, ,
t. ?
, p
g:E%-.1 ,..;
4) '5x E4)
E-44-4)
Uranium
content
in melt,
wt.%
Contact
angle of
wetting.
deg
2
Iron
11,1
1250
0,12
68?5
3
Thesame
19,2
1250
0,18
75?5
7
Nickel
20,2
1300
1,92
44?4
8
..
30,3
1300
2,20
37?5
.3
Copper
20,0
1200
0,48
62?5
12
Steel
4,9
1300
0,92
"71?5
Khl8N10
Fig. 1. External appearance of alloys with additives: 1) 10% Si; 2) 20.2% Ni; 3) 30.1%
Ni; 4) 49.2% Fe; 5) 29.4% Fe; 6) 46.3% Cu:
Translated from Atomnaya gnergiya, Vol. 32, No.6, pp.490-491, June, 1972. Original article sub-
mitted August 12, 1971.
C 1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York,
N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without
permission of the publisher. A copy of this article is available from the publisher for $15.00.
579
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M1 copper, Ni nickel, and Kh18N10T stainless steel. We studied the conditions for the alloying of zir-
conium with these metals in the phase interval bounded by the points of intersection of the liquidus line with
the 1300?C isotherm. The alloying of zirconium with the metals and the interaction of the melts with UO2
were carried out in crucibles of transparent quartz in an induction-type vacuum furnace. The temperature
was measured with an optical pyrometer. In studying the wettability of uranium dioxide by the melts, we
used the stationary-drop method. The experiments were conducted in a resistance furnace at a vacuum of
5.10-2 mm Hg.
When we heated zirconium specimens with metals, the liquid phase was formed as a result of contact
melting in every experiment. The zirconium alloying process is fastest (10-20 sec) with copper, which may
be explained by the absence of high-melting compounds in this system. At temperatures below i300?C,
specimens containing 11-30% iron, 12-30% nickel, or 5-28% stainless steel (see Fig. 1) become completely
melted.
The resulting zirconium alloys have high fluidity and low surface tension, and therefore they spread
out over the bottom of the crucible.
The zirconium alloys wet the uranium dioxide to different degrees. The contact angle of wetting is
largest for zirconium?iron alloys [(68-75) ? 51 (see Table 1). At 1250-1300?C there is a slight interaction
of the zirconium in the melt with the solid uranium dioxide: Zr + UO2 = Zr02 + U. The reduced uranium is
dissolved in the melt. The amount of uranium going into the melt is largest in the case of zirconium alloyed
with nickel (2.2%) and stainless steel (0.92%) and smallest in the case of alloys with iron (0.18%) and copper
(0.48%).
Thus, we have experimentally established the conditions for the alloying of zirconium with a number
of metals at temperatures ranging up to 1300?C. We have shown the absence of any substantial interaction
or wettability of the uranium dioxide with the melts containing zirconium and the specified additives. The
resulting data make it possible to undertake research on the thermal breaching of fuel elements with zir-
conium jackets by melting the jackets in the presence of metallic additives.
The authors wish to express their gratitude to N.A.Nilov for his interest and assistance in connection
with the work, as well as to E.M.Valuev and E. V.Komarov for their help in carrying out the experiments.
LITERATURE CITED
1. G.P.Novoselov and A.T.Ageenkov, Atomnaya Energiya, Vol.16, 230 (1969).
2. E.Hayes et al., Trans.ASM, Vol.43, 888 (1951).
3. C. Lundin et al, J.Metals, Vol.5, 273 (1953).
4. B. Lustman and F.Kerze, The Metallurgy of Zirconium [Russian translation], IL, Moscow (1959).
580
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2104
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X-RAY FLUORESCENCE ANALYSIS OF HEAVY
ELEMENTS BY aX-COINCIDENCE
B. M. Aleksandrov, S. M. Solov'ev,
L. I. Tyvin, and V. P. Eismont
UDC 543.53
One of the important methods of radioisotope analysis of heavy elements is the a-spectrometer. At
the present time, about 1000 a-lines of 40 natural and 240 artifical a-emitters are known [11. However,
more than 90% of these lines are concentrated in a narrow energy range (4.5-8.0 MeV). Therefore, it is
natural that the a-lines of many elements appear very close to one another and cannot be resolved by semi-
conductor a-spectrometers applicable to isotope analysis, the better models of which have a 10-15 keV re-
solution. At the same time, one can only obtain such resolution with very narrow sources.
The a-, y-coincidence appears to be ineffective in several cases; for example, for even?even nuclei.
All are characterized by the rotational nature of the low-lying (the most intensively populated) excitation
levels whose energies differ little from nucleus to nucleus. Aside from this, transitions from these levels
are highly converted (the internal conversion coefficient ?1000) so that they are relegated to a very small
part of the y-radiation. However, as a result of the internal conversion, there arises an intense x-ray L-
radiation with the transition on the order of 0,1 quantum per a-decay. On the energy distribution curve for
the fluorescent L-radiation obtained by ionization processes, there are three high spikes La, Li3, and Ly,
whose energies increase regularly with an increase in the nuclear charge Z shifting with AZ = 1 by roughly
300, 500, and 700 eV, respectively [2]. This shift appears sufficient for definite separation of adjacent ele-
ments with respect to Z, if one utilizes up-to-date semiconductor Si(Li)- or Ge(Li)-spectrometers whose
energy resolution is several hundred electron volts for measurement of the spectra. The limitation of the
x-ray fluorescence technique, associated with the difficulties of its utilization for the analysis of the isotopes
100
90 ?
80 ?
70 ?
10
Et
15
Fig. 1
211
25
Ex, keV
Am 2"
effi242 Cf 252
c''"
,?, 243,244
. ?
?1?107"1
5,2
s 6
-;
Fig. 2
5,0
5,0
Ea , MeV
Fig. 1. Dependence of the response et of the spectrometer on the radiation energy (e is
the sensitivity; t is the penetration factor).
Fig.2. The a-particle spectrum for the target being analyzed.
Translated from Atomnaya Energiya, Vol.32, No.6, pp.491-493, June, 1972. Original article sub-
mitted August 17, 1971.
o 1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York,
N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without
permission of the publisher. A copy of this article is available from the publisher for 815.00.
581
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Declassified and Approved For Release 2013/03/01 : CIA-RDP10-02196R000300100005-6
.
.
.
04%,....,
1
:
.
.
.
quo'
.:.
.
?
.
a
:????%.?%.,
.i ..m.,?,?,
.. . .:
....
. ?
....
:v.:, ?
.
?
..
. .
b
:.
...
? .?
? ??? ??...
???...._.,,d.......*
C
.?
. ..
. ?
:
'3,........:.??-? ?:???????????
.*?'
Pf" ? ?r?
.
.
.
.
..
.?
?
?
"v?
.
, ?
.,...4....../
.r.
.
'..
:"
. .
. .
.
.
.
?
. .
d
. 10....
, ? ..
????........, %.,.....
:.?
?
--
?
/
i.
1
1
1 ? .
A? ?
/ '
..
/
/
1
e
I ???
? .
?
/ - ____:',.........
/2
16
20 E keV 24
Fig. 3. Spectra of char acteris tic L-radia-
tion accompanying a-decay: Am241 (a);
cra242 (b); cf252 (c); Am241 cm242-244 cf252
isotope mixture (d); and CM242 + Cf252 mixture
(e) . The individual spectra for Cm242 and cf252
(in the ratio ,found through the number of
spontaneous fissions), as well as their
sum, are shown in the last figure.
of a single element, appears to be unimportant mostly due to
the well-known regularities of the a-decay coinciding with
the a-spectra of nuclei differing with respect to A and Z. For
example, the lines of PU238 (5499 keV, 71%; 5456 keV, 29%)
and Am". (5486 keV, 86%; 5443 keV, 13%); as well as Cm242
(6112 keV, 74%; 6068 keV, 26%) and Cf282 (6118 keV, 84%;
6075 keV, 16%) are similar.* Apart from this, with absence
of energy shifts in the L-radiation (in isotopes of a single
element), analysis with respect to the number of quanta of
L-radiation per decay event, which can be different for dif-
ferent isotopes 'is possible. These considerations also led
to development of the aX-coincidence method for the analysis
of heavy elements.
Surface-barrier Si(Au)- and diffusion-drift Si(Li)-de-
tectors are utilized for the measurement of a-particle and
x-ray quanta energy distributions, respectively. They were
enclosed in a single vacuum chamber into which a cold-duct
was introduced. In the cold-duct, which is lowered into a
Dewar of liquid nitrogen, there were mounted an x-ray de-
tector and a field transistor [3]. Above the detector, a
covered, 0.1 mm thick, beryllium disk, there is located, at
a distance of several millimeters from gold foil, the target
being analyzed, above which, in its turn, there is arranged
an a-radiation detector. The energy resolution is about 700
eV for the x-ray channel, and 80 eV for the a-particle chan-
nel. The resolving time for the coincidence circuit 2r rzl 1
psec and cannot be substantially reduced because the life-
time of the low-lying excited levels of several nuclei, for
example, the 59.6 keV level of Am241, amounts to several
times 10-8 sec.
Semiconductor detectors of electromagnetic radiation
require sensitivity calibration and determination of its depen-
dence on the energy of the radiation. For this, the widths of
the "window" and of the operation area are measured. In the
present work, these quantities were determined for the rela-
tive intensities of x-ray and soft y-radiation lines in Cou
and Aram spectra, which were shown to be equal to 0.007
and 2.45 mm, respectively. The energy response curve a
for the x-ray channel allowing for absorption in the target
backing, the beryllium disk, and the window of the counter
is shown in Fig. 1.
In the capacity of a control model, we solved the prob-
lem of determining a small amount of californium in a target
of Alum cm242-244 cf252. As has already been mentioned,
the a-lines of Cf282 and Cm242 are similar to one another, and
a direct determination of Cf282 through the a-spectrum is dif-
ficult. The spectrum of the target being analyzed is shown in
Fig.2. The prominent lines are Am241, cm2 43, 244, and Cm242
+ Cf282; the last also includes several transitions to the lower
levels of Cm243: their contributions to this line comprises
several percent. With utilization of the aX-coincidence meth-
od for the quantitative analysis of the relative contributions of curium and californium to the overall a-line,
it is assumed that the relative yields in x-rays per a-decay event are well-known for these elements. For
the determination of the relative yields of radiation for Cm242 and Cf282, there were prepared two additional
*The relative intensities of the lines are indicated by the percentages.
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targets: Am241 Cm242 and Am241 Cf252. All three targets were prepared by the technique of thermal pul-
verization in a vacuum of a blended mixture of the respective isotopes that by itself did not only attain a
high uniformity, but also identical surface distributions of curium, californium, and americium. This al-
lowed us to utilize the americium as a reference for each of our targets. All of the measurements for each
of the targets were also carried out relative to this reference by means of a step-by-step separation of the
corresponding a-lines (with the aid of a differential a-channel discriminator). The spectrum of the L-radia-
tion, corresponding to this line, was measured, and the number of aL-coincidences per a-particle NaL/Na
for americium and the respective isotopes (or their mixture) was determined. In this way, the relative
yields of L-radiation per a-decay were determined for Cm242,
Cf252:Cf252, and Cm242
66/'.= ? 6N" = ; and 6N"'?
N?L/Na NaL/Nc, NaL/Nm ?
In each case, more than 10,000 coincidences were registered and it was found that ON = 0.310, ON" = 0.147,
and or" = 0.277. The background level was high only for Cm242 + Cf252, where it comprised about 50% of the
total number of coincidences. However, the background had to be determined with a specified precision, so
that the relative error in the value of ON did not exceed 2%. Denoting through p that part of the a-activity
of Cf252 in common with the a-lines of Cm242, it is not difficult to obtain the ratio p = ? 6N")/6N, ? ON"
and one finds that p = 0.20 ? 0.05. (In addition, the contribution of the californium's activity to the total a-
activity of the target equals 0.016 ? 0.002.)
New information on the composition of the isotope and its atomic number (when the composition of the
elements is unknown), can be obtained from the L-radiation spectra. The energy distribution of the L-series
of the characteristic radiation, obtained by us, during a-decay of Am241 (a), Cm242 with the admixture of
Cm243 mentioned previously (b), Cf252 (c) as well as L-radiation spectra of the target being analyzed without
coincidence with some a-line (d), and under conditions involving coincidence with an a-line of Cm242 Cf252
(e) are presented in Fig. 3. It is seen that an overwhelming part of the target's total x-ray spectrum is due
to the americium. In the same spectrum, corresponding to the a-line of Cm + Cf, the contribution of the
californium is already found to be pronounced.
In the present work, the Cm242 + Cf252 combination was not selected by chance. The californium iso-
tope has a high rate of spontaneous fission that allows an independent estimate of its content in a target.
It was found that the fraction of a-activity p for californium in the Cm + Cf line is 0.26 ? 0.02 which, within
the limits of error in the determination, coincides with the value cited above. The value (ON'/ON") = 2.11
? 0.06, therefore, the contribution of californium to the L-radiation of the two isotopes, calculated in this
way, equals 0.15 ? 0.02 (with a total L-radiation for the target of 0.0033 ? 0.0004). For comparable cross
sections with the ratio (1-0.15)/0.15, the distributions for Cm242 and Cf252 are added. These components and
the results of their addition are shown in Fig. 3e, The comparison shows that the measured L-radiation dis-
tribution for a mixture of these isotopes agrees with the theoretical distribution. Some broadening in the
spectrum of the mixture in comparison with the sum of the spectra for the individual components is associa-
ted with the large, in this case, load of the x-ray channel, rather than with measurements of the spectra for the
separate isotopes.
In our opinion, the example presented corroborates the possiblity and feasibility for the utilization of
a semiconductor aX-coincidence spectrometer for the analysis of a complex mixture of the heavy elements.
The authors thank N.A.Perfilov for interest in the work,
LITERATURE CITED
1. J.Post, Actinides Rev., 1, 55 (1967).
2. E.Storm and H.Israel, INiucl.Data Tables, A7 (1970), p.565.
3. S.M.Solov'ev et al., Pribory i Tekhnika asperimenta, No.1 (1972).
583
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EFFECTIVE NEUTRON ABSORPTION CROSS SECTIONS
FOR Cf252 AND Cf253 IN THE CENTRAL CHANNEL OF
AN SM-2 REACTOR
V. A. Anufriev, V. D. Gavrilov,
Yu. S. Zamyatnin, V. V. Ivanenko,
and G. N. Yakovlev
UDC 539.125.5.173
The effective cross sections for absorption of neutrons by californium isotopes, measured for various
reactors, differ from each other by an order of magnitude, which is significantly greater than the experi-
mental error [1, 2].
The variation in the characteristics of reactor neutron spectra, which is one of the causes of the ob-
served spread, limits the possibility of using the cross sections in the calculation of isotope-accumulation
processes in inhomogeneous reactors.
To determine the effective neutron-absorption coefficients in the central channel of the SM-2 reactor,
we irradiated two targets consisting of Cf252 with an impurity of a lighter californium isotope (-20%). The
isotope composition of the irradiated targets was investigated on a semiconductor a-spectrometer having
an energy resolution?-?:125-30 keV. The integrated neutron fluxes measured by cobalt monitors were 1.8.1021
and 1.1-1021 neutrons/cm2. The Cf252 depletion cross section was found to be 72 ? 18 barn. Keeping in
mind the theoretical estimates for the Cf252 fission cross sections [3], one may attribute the measured value
completely to the value of the effective radiation-capture cross section. For Cf253, whose content in the
samples after irradiation was determined by the accumulation of daughter ES253, the effective absorption
cross section was found to be 6260 ? 1800 barn. Due to the smallness of the integrated fluxes, the quantita-
tive Cf254 content in the irradiated samples can be estimated only approximately. Assuming that the deple-
tion cross section of Cf254 is -100 barn, the Cf253 radiation-capture cross section is estimated to be gzi10
barn.
The magnitudes for the cross sections which we have found in this work are significantly larger than
the published findings [1, 2, 4]. A possible reason for this is the hardness of the neutron spectrum of the
SM-2 reactor. The ratio of the fission and radiation-capture cross sections for Cf253 is significantly greater
than the analogous ratios for other transuranic nuclei and it limits the possibility of Cf254 accumulation. In-
asmuch as the Cf253 depletion probability is much greater than its 13-decay probability, the optimal regime
for obtaining einsteinium isotopes appears to be the alternation of cycles of irradiation with extraction which
allows transformation of accumulated Cf253 into ES253,
LITERATURE CITED
1. E.Hyde et al., Nuclear Properties of the Heavy Elements [Russian translation], Vol.2, Atomizdat,
Moscow (1968), p.61.
2. I.Halperin, ORNL-4306 (1968).
3. A, Prince, in: Californium-252, Proc. of Symposium, CONF-681032, Sponsored by the New York
Metropolian Section of the American Nuclear Society, New York (1968), p.23.
4. I.Halperin, Nucl.Sci. and Engng., 37, 228 (1969).
Translated from Atomnaya Energiya, Vol.32, No. 6, pp.493-494, June, 1972. Original article sub-
mitted August 20, 1971.
584
o 1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York,
N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without
permission of the publisher. A copy of this article is available from the publisher for $15.00. .
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Declassified and Approved For Release 2013/03/01 : CIA-RDP10-02196R000300100005-6
UNFOLDING FAST-NEUTRON SPECTRA FROM
THRESHOLD DETECTOR ACTIVITIES
A. M. Aglitskii, S. S. Lomakin,
A. G. Morozov, V.1. Petrov,
and S. G. Popov
UDC 621.039.512.45
Integral fast-neutron spectra in the experimental channels of the reactor at the Novo-Voronezh Atomic
Power Station were measured by the threshold detector method and the values of the activation integrals
were determined for each detector [1].
In this paper, differential fast-neutron spectra are unfolded on the basis of the values of the activation
integrals found in [1]. The Dierckx method, the iteration method [2], a rapid method [3], and the spectral
indices method [4] were used to unfold the spectra. The essence of all these methods is the following.
4,(E), rel. units
/02
101
lo?
tot-0
:2
(p(E), rel. units
10?
leo
lot-4)
2
?
0 1 2 3 4 5 6 7 8 9 10 1
E,MeV E, MeV
Fig. 1 Fig. 2
Fig. 1. Fast-neutron spectra at the center of the reactor core: 1) neutron spectrum un-
folded by the rapid method; 2) fission-neutron spectrum; 3) neutron spectrum unfolded
by the method of spectral indices.
Fig. 2. Fast-neutron spectra inside (a) and outside (b) the reactor vessel as unfolded:
1) by the iteration method; 2) by the rapid method; 3) by Dierckx method.
2 3 5 6 7 8 9 10
a
Translated from Atomnaya Pnergiya, Vol.32, No.6, pp. 494-495, June, 1972. Original article sub-
mitted September 2, 1971.
C 1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York,
N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without
permission of the publisher. A copy of this article is available from the publisher for $15.00.
585
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The specific activity A of irradiated threshold detectors (activation integral) is related to the threshold
reaction cross section o(E) and the desired neutron spectrum (ME) by an integral relation of the form
Ai= $
cri(E)ip (E) dE, i = 1, 2, 3, n, (1)
0
where n is the number of threshold detectors. For a unique solution of this equation system, it is necessary
to make some assumption about the mathematical form of the desired spectrum co(E), i.e., it is necessary
to represent spectra in the form of expressions with a finite number of unknown parameters.
In the first three methods, the energy range under consideration is broken up into intervals in each of
which the desired spectrum is in the form
(E)= C exp (? RE). (2)
The boundaries of the intervals are determined by the threshold detector sensitivity functions cri(E)(p(E).
The unknown parameters C and ? in each interval are determined from the continuity condition for the spec-
trum and from the equations obtained when Eq. (2) is substituted into Eq. (1). It is necessary to have two
threshold detectors in the last interval. The activation integrals Ai are used as input data in Diercloct meth-
od; in the rapid and iteration methods, the integral fast neutron fluxes determined by the effective threshold
cross section method are used. The effective threshold energy Eieff and the effective threshold cross sec-
tion ai are introduced in the following manner;
eff
Ai= ai (E) p (E) dE =ei ff (E) dE = criff(1) (Eei ff),
E2eff
(3)
where .1.(Eeiff) is the integral flux of neutrons with energies greater than Eei ff.
In the rapid method, crieff and Eieff are determined from several known spectra [3]. In the iteration
method, these quantities are determined from the fission-neutron spectrum in accordance with Grundl and
Usner [5]. Using these values, one unfolds a desired neutron spectrum from which new values of Eel ff are
determined (crlff retains its previous value). Then the procedure is repeated.
e
In all the methods mentioned above, the unfolding process starts with the last interval.
The method of spectral indices is used in the case where the desired spectrum is not much different
from the fission-neutron spectrum. Such a spectrum can be represented in the form
(E)= C El I 2 exp (? 13E) (4)
(for a fission spectrum = 0.775). Values of the activation integrals
CO
i= CE1/2 exp (? 13E) ai (E) dE
0
(5)
are calculated for various
By comparison of the experimental quantities Ai/Ai and the calculated A'iflij, that value of /3 is selected
for which the difference between these ratios is a minimum.
Fast-neutron spectra unfolded by these methods are shown in Figs.1 and 2.
On the basis of these results, one can arrive at the following conclusions; 1) the neutron spectrum at
the center of the core is similar to a fission-neutron spectrum; 2) the neutron spectrum inside the reactor
vessel has a shape characteristic of spectra from neutrons that have penetrated a layer of water; 3) the
neutron spectrum outside the reactor vessel is "softer" than the neutron spectrum inside the vessel; 4) the
neutron spectra obtained by the various methods are in satisfactory agreement. The greatest difference is
observed in the low-energy region.
LITERATURE CITED
1. S.S.Lomakin et al., At.Energ., 31, 54 (1971).
2. Neutron Dosimetry, IAEA, Vienna (1963), pp.27, 325.
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3. V. S.Troshin and E.A.Kramer-Ageev, At. Energ., 29, 37 (1970).
4. K.Beckurts and K.Wirtz, Neutron Physics [Russian translation], Atomizdat, Moscow (1967).
5. W. Zijp, Review of Activation Methods for the Determination of Fast Neutron Spectra Pattern (May,
1965), p.45.
587
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ANGULAR DISTRIBUTIONS OF PHOTONEUTRONS
FROM Al, Ti, Cu, Mo, W, AND Pb
V. P. Kovalev, V. P. Kharin,
V. V. Gordeev, and V. I. Isaev
UDC 539.125.5.18
Information on the angular distributions of photoneutrons for energies of excitation of the nucleus of
15 to 30 MeV is important for clarifying the mechanism of photonuclear reactions in the region of the giant
dipole and quadrupole resonances.
In most studies of the angular distributions of photoneutrons [1-4] targets were irradiated by colli-
mated beams of 'y-rays of various energies. Considerably less effort has been devoted to the study of the
angular distributions of photoneutrons from various targets bombarded directly with electrons, although
in most practical applications just such a method is used to obtain photoneutrons [5].
It is shown in [61 that the cross section for the electron disintegration of a nucleus in the 13-17.5 MeV
range is about 400 times smaller than the corresponding cross section for photodisintegration, but the en-
ergy thresholds of the reactions are the same. One can conclude from this that for electron energies of
?20 MeV most of the photoneutrons are produced in (y, n) reactions.
TABLE 1. Angular Distributions of Photoneutrons
1Target
lElectron
energy, MeV
P.:,
2
0
M
Angle, deg
B/A
30
50
70
90
110
130
150
Al
22,5
P31(n,p)
1,05+0,08
1,03+0,08
0,97+0,08
1,0+0,08
0,98+0,08
1,02+0,08
1,04+0,08
Isotropic
22,5
A127(n,p)
0,90+0,15
0,95+0,15
1,02+0,15
1,00+0,14
0,96+0,13
1,07+0,13
1,01+0,13
.,
Ti
22,5
P31(n,p)
1,04+0,07
0,96+0,07
1,03+0,07
1,00+0,07
0,98+0,07
1,05+0,07
1,03+0,07
.
22,5
A127(n,p)
1,06+0,13
0,94+0,13
1,04?0,121,00?0,12
0,95+0,11
0,98+0,11
1,02+0,10
.
Cu
12,8 11131 (n,p)
0,97+0,10
1,04+0,10
1,02?0,1014,00?0,10
1,01+0,10
0,90+0,10
0,96+0,10 .
17,0
P31(n,p)
1,03+0,07
0,97+0,07
1,00+0,0711,00+0,07
1,06+0,07
0,95+0,07
0,88+0,07
22,5
P31 (n, p)
0,87+0,05
0,94+0,05
0,97+0,05
1,00+0,05
0,99+0,05
0,93+0,05
0,91?0,050,18?0,04
22,5
A127(n,p)
0,75+0,09
0,86+0,07
0,98+0,06
1,00+0,05
1,02+0,05
0,94+0,04
0,90?0,040,28?0,06
Mo
22,5
P31(n,p)
0,90+0,05
0,93+0,05
0,98+0,05
1,00+0,05
0,99+0,05
0,92+0,05
0,84?0,050,21?0,04
22,5
A127(n,p)
0,80+0,08
0,95+0,08
0,95+0,07
1,00+0,06
0,94+0,05
0,83+0,04
0,72?0,040,44?0,01
22,5
A127(n,a)
0,72+0,08
0,84+0,08
0,89+0,08
1,00+0,08
0,95+0,08
0,87+0,08
0,63?0,080,78?0,11
W
22,5
P31(n,p)
0,85+0,04
0,90+0,04
0,98+0,04
1,00+0,04
0,98+0,04
0,92+0,04
0,087?0,040,25?0,04
22,5
A127(n,p)
0,78+0,06
0,84+0,06
0,89+0,05
1,00+0,05
0,97+0,04
0,86+0,04
0,75?0,040,54?0,06
Pb
22,5
P31(n,p)
0,79+0,04
0,85+0,04
0,96+0,04
1,00+0,04
0,98+0,04
0,88+0,04
0,84?0,040,36?0,02
22,5
A127(n,p)
0,70+0,09
0,81+0,08
0,94+0,07
1,00+0,06
0,94+0,06
0,80+0,05
0,69?0,050,69?0,12
Translated from Atomnaya nergiya, Vo1.32, No.6, pi:1.496-497, June, 1972. Original article sub-
mitted September 6, 1971.
588
1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York,
N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without
permission of the publisher. A copy of this article is available from the publisher for $15.00.
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Neutron yield, rel. units
Declassified and Approved For Release 2013/03/01 : CIA-RDP10-02196R000300100005-6
1,2
1,0
0,8
1,2
1,0
Pb
[
0,8
-f
1,2
1,0
Mn
0,8
1,2
1,0
Cu
--
1
0,8
1,2
Ti
1,0
f
0,8
1,0
48 1,2
_}At
I
II
f_
30 50 70 90 110 130 150
e,deg
Fig. 1. Angular distributions of fast photo-
neutrons from Al, Ti, Cu, Mg, W, and Pb
bombarded with 22.5 MeV electrons. P31
(n, p) Si31 detector.
We have studied the angular distributions of photo-
neutrons from Al, Ti, Cu, Mo, W, and Pb targets bombarded
with monoenergetic electrons having energies from 12 to 23
MeV. The targets were cylindrical in form and 15 mm in
diameter. The target thicknesses were as follows:
Element
? Thickness,
g/cm2 4.05 6.75 13.32 15.54 19.3 17.0
The threshold reactions 1331(n, p) SaI, A127 (n, p)Mg27,
A127 (n, a)Na24 with effective thresholds of 2.7, 4.5, and 8.1
MeV , respectively ,were used for detection. The experiments
were performed with a linear electron accelerator beam
having a maximum energy of 25 MeV [7]. An electron beam
?-?10 mm in diameter was directed onto the face of a target
placed 30 cm from the exit window of the accelerator. The
neutron detectors were placed around the target at fixed
angles. The angles subtended by the detectors were 4 and
60, respectively ,for phosphorus and aluminum detectors. The
activity induced as a result of the 1331(n, p) Sin reaction was
recorded by a gas counter combined with standard counting
equipment. The A127 (n, p)Mg27 and A127 (n, a) Na24 reactions
were detected by their characteristic y-radiation using a
single-crystal y-spectrometer. . The results of the measure-
Al Ti Cu Mo W Pb
ments with the P31 (n, p) detector and an electron energy of 22.5 MeV are shown in Fig. 1. The a Lr dis-
tributions of the fast photoneutrons are approximated by the relation
f (0) ==i1?Bsin20.
(I)
Analysis of the experimental data shows that for light elements (Al, Ti) the angular distributions are
isotropic within the limits of statistical error. As the atomic number increases the angular distributions
tend to become more anisotropic around 90?. The ratio 13/A of Eq. (1) is used as an anisotropy index. The
anisotropy also increases with increasing threshold for detecting neutrons.
Table 1 shows the results of the measurements of the angular distributions of photoneutrons. They
are presented as yields relative to the yield at 90?. The ratio B/A is given as a function of electron energy
and the threshold energy for detecting neutrons.
The shape of the angular distributions of photoneutrons for elements of medium and high atomic num-
bers indicates the dipole character of the absorption of y-rays in the energy range studied. No appreciable
asymmetry in the angular distributions of photoneutrons is observed at these excitation energies.
LITERATURE CITED
1, L. E. Lazareva and A.I.Lepestkin, Yadernaya Fizika, 11, 266 (1970).
2. F.Ferero et al., Nuovo Cimento, 4, 2, 418 (1956).
3. K.McNeill et al., Canad.J.Phys.,-46, 1974 (1968).
4. Hiroaki Tsubota et al., J.Phys.Soc.Japan, 26, 1 (1969).
5. D. Gayther and P. Goode, J.Nucl.Energy, 21,733 (1967).
6. L.Skaggs et al., Phys.Rev., 73,420 (1948).
7. V. I.Ermakov et al., Atomnaya Energiya, 29, 206 (1970).
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Declassified and Approved For Release 2013/03/01 : CIA-RDP10-02196R000300100005-6
CALCULATION OF THE BOUNDARY EFFECT IN
PARTICLE-TRANSPORT PROBLEMS
S. I. Durinov, V. V. Uchaikin, UDC 539.12.172
and A. M. Koltchuzhkin
The particle flux at a medium-vacuum boundary for a plane-parallel geometry can be expressed in
terms of the particle flux in a homogeneous infinite medium (which is an easier calculation to make), and
the differential albedo -whose numerical values are given in the literature for a number of cases.
We introduce the following notation: co and E are the direction and energy of the moving particles,
and j(z, co, E) is the differential particle flux along the z axis, which is related to the differential flux ,D by
j(z, o, E)= cos GO (z, o, E).
For definiteness, we assume that the particle source is to the left of the observation point z. Evidently,
the particle flux ji that intersects the density z.= const for the first time equals the particle flux under the
boundary-geometry conditions (when the medium-vacuum boundary passes through the point z).
In an infinite medium, a fraction of these particles are reflected from the right with respect to the
point z of the half-space and generate a flux j2 in the half-space, which is related to ji by
f2(z, (0, E)-=iffi(z,
where
!III III(
?32 44 0,6 0,8 0
E, MeV
dE'a(u, Eke, E')
r-
30? 600 900 120? 150? 1800
(180?- 0?)
Fig. 1 Fig. 2
Fig. 1. Effect of boundary on energy distribution of quanta (110z = 4): ) energy dis-
tribution of quanta in infinite medium; ----) energy distribution of quanta at the
medium-vacuum boundary in the ji = j?; -?-? -?) energy distribution of quanta at the
medium-vacuum boundary, obtained from Eq. (7).
Fig.2. Effect of boundary on angular distribution of quanta (ttoz = 4).-) angular
distribution of quanta in infinite medium; ----) angular distribution of quanta at the
medium-vacuum boundary, obtained from Eq. (7).
Translated from Atomnaya Energiya, Vol.32, No.6, pp.497-499, June, 1972. Original article sub-
mitted October 7, 1971.
590
0 1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York,
N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without
permission of the publisher. A copy of this article is available from the publisher for $15.00.
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Declassified and Approved For Release 2013/03/01 : CIA-RDP10-02196R000300100005-6
TABLE 1. Dose Buildup Factors B for an
Infinite Medium and at the Medium-Vacu-
um Boundary
cos 00=1
CO8 00=0 , 75
cos 80=0 ,5
1.4?z
B
B+
Bi
B
B+
Ili
B
13+
Ba
0,5
1,57
1,45
1,42
1,68
1,51
1,47
1,82
1,62
1,57
1,0
2,03
1,81
1,77
2,24
1,99
1,92
2,61
2,29
2,20
2,0
2,98
2,64
2,57
3,63
3,14
3,09
5,40
4,79
4,69
4,0
5,10
4,48
4,39
8,85
7,87
7,71
27,4
24,1
23,7
Note: 131 are exact values B+ are values from the
j j+ approximation.
is the integral operator of the albedo, whose kernel is the
differential current albedo a(co, Elcot, Et), where
a (co, E I co', E')-= 0, cos 0' cos 0 >0. (1)
As a result of the reflection of the particle flux j2 from
the left half-space, a flux
/3 (z, (0, E)= Aj 2 (z, co, E)
is generated, etc. For the jn-th flux, we have the recurrence
relation
/0+j (z, (0, E) Ajn (z, (o, E),
and the sum of all these fluxes is equal to the particle flux in the infinite medium:
CO
j (z, o, E)=-- E n (z, c), E),
n=i
which we assume to be known. Note that
j 2? (z, co, E) =0, for cos 0 > 0;
/1(z, (0, E)=0, for cos 0 < O. j
Substituting Eq. (2) into (3) and summing the operator series, we obtain the relation
Jj (z, co, E)== j (z, co, E)- Aj (z, co, E), (5)
which relates the radiation field at the boundary and the field in the infinite medium. This equation is valid
for any source with plane symmetry.
We introduce the notation
(2)
(3)
(4)
{ /(z, co, E), cos 0 > 0;
co, E) =
0, cos 0 < 0; (6)
0, cos 0 > o;
i_ (z, (0, E)= f
j (z, co, E), cos 8 < o.
Assuming in Eq. (5) first cos 8> 0, and then cos 0 < 0, and taking account of (1), (4), and (6), we obtain
j (z, co, E)= 1+ (z, (0, E)- Aj_ (z, o, E);
j_ (z, co, E)= Aj+ (z, 0), E),
whence
(7)
j (z, w, E)=- [1 - I+ (z, (0, E). (8)
For 112211 ? 1, the second term in Eq. (8) can be neglected, and we can assume
ji (z, E) j+ (z, co, E).
This approximation was used by Fano et al. [11 to estimate the boundary effect in the theory of y-radiation
transport.
A numerical example illustrating Eqs. (7) is given in Fig. 1 for a '-quanta flux in a plane-perpendicular
source with E0 = 1 MeV in aluminum. From Fig.1 we can see that the boundary effect is most important in
the low-energy part of the spectrum, where the correction due to the second term in (7) can be considerable.
Figure 2 shows the angular distribution of the photon flux obtained from (7). The figure shows that the
difference between the fluxes in the infinite medium and at the boundary depends weakly on 0 for 0 =7r/2.
This result is related to the fact that the boundary affects primarily the low-energy part of the spectrum,
which has an angular distribution that is nearly isotropic.
If the detector recording the radiation flux has a sensitivity function p(co, E), then for the functional
J (z)= do) dE p (co, E) see Oj (z, co, E)
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from Eqs. (7) we obtain
where
J (z) = .1+ (z) ? do)' dE' j_ (z, c)' , E') sec 0' p (o', E') A (w', El)
I
A (co' , E')= do) dE a (o), E I co' , ) coo 0' P (co, E)
cos 0 I p(0)',E')
(9)
is the integrated albedo [21.
Table 1 shows the results of a calculation based on Eq. (9) for the dose buildup factor for concrete
layers for various incident angles of primary y-quanta with E0 = 1 MeV. In the present study the differential
particle flux in the infinite medium was calculated by the Monte-Carlo method, and the data on the albedo
was taken from [3]. The calculated values of the buildup factors are in good agreement with the results
from a direct calculation by the Monte-Carlo method [3].
Equations (7) and (9) can be used in problems with neutrons, 'y-quanta, electrons, and other particles,
if their differential fluxes in an infinite medium and integrated albedo are known.
LITERATURE CITED
1. U.Fano, L.Spencer, and M.Berger, y-Radiation Transport [Russian translation], Atomizdat, Moscow
(1963).
2, B. P.Bulatov et al., y-Radiation Albedo [in Russian], Atomizdat, Moscow (1968).
3. D.J.Raso, Nucl.Sci.and Engng., 17, 411 (1963).
592
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METHODS FOR CALCULATING NONSTATIONARY
PROCESSES INVOLVING SEPARATION OF MULTICOMPONENT
ISOTOPIC MIXTURES IN PACKED COLUMNS
I. A. Vereninov and Yu. V. Rakitskii UDC 621.039.31
The use of a digital computer for investigating separation processes requires the development of a
mathematical model whose form is capable of being realized in the computer.
Consider a rectangular-multistage system used in two-phase separation of an m-component mixture
without isotope exchange. A mathematical model which has been discussed previously [1, 21 may be de-
scribed by a set of partial differential equations in the joint domain satisfying the following conditions at
the section boundaries:
axl writ axf AraGs s s Gs 00.
ns+ Ls at Ls
al (1 ? - - eq,4;
r=1
G2yt. (0, 0+ F y (F --=Glyt (1, t);
dxf (1, t)
4 dt (La? Ls+1) [xl, (1, t)? ylk(t)];
Gsyl (1, t)=G8i-lysi+1 (0, 0+ (Ls ? Ls+1) yfh (t);
dx1 (0 , t)
dt
iyit 0, ?xi 0].
(1)
Here the superscript s denotes the section number (s = 1,2,3, ,b); the subscript i denotes the compo-
nent number; t is a dimensionless coordinate which varies from 0 to 1 along the direction of motion of the
fluid; xi and yi are the concentrations of the i-th component in
the liquid and gas currents; L, G, and F are the liquid, gas, and
supply currents; S2 is the liquid holdup; H is the height of a sec-
tion; N is the number of stages; EK is the quantity of liquid in the
phase return systems; and eici is the enrichment factor for the
i?q isotope pair.
In order to solve (1) it is necessary to convert it to a set of
difference equations, but in this case we do not get satisfactory
speed and accuracy of solution by straightforward mesh and col-
location methods on account of the details of the processes con-
sidered, which are reflected in (1) and represent a two-dimension-
al boundary layer with the parameters N, 2, EK, G, and L. To
construct the physical and mathematical basis of a different model,
the following are therefore necessary:
to use an integral form for (1);
to carry out asymptotic transformations with respect to t
and the functions y( S, t) of the variable ts for the bound-
ary conditions in the phase transformation systems [3];
to make the most accurate approximation the asymptotic
form for ys(ts' t)
i
4
1,0
0,9
0,8
47
46
0,5
44
43
42
41
4.5
43
0 8 16 24 32 40 48 55 64 72 80 88 95
t, days ?
Fig. 1. Curves for attainment of the
steady state by a two-section stage
for an average component x (set pi,
is used for t = 0 and set p2 is used
for t = 30 days, where Mi = M2 = 5).
Translated from Atomnaya Energiya, Vol.32, No.6, pp.499-501, June, 1972. Original article sub-
mitted October 7, 1971.
41) 1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York,
N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without
permission of the publisher. A copy of this article is available from the publisher for $15.00.
593
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Declassified and Approved For Release 2013/03/01 : CIA-RDP10-02196R000300100005-6
We can use the values and derivatives of the variables at the approximation points, so it is best to use
Hermite polynomials for the xis. (t s, t) [4].
The differential-difference model was constructed by taking for each section Ms points with steps of
hs = ? tg, where n is the number of the point.
We get from asymptotic transformation at selected points the. following forms for (1), the integral
forms of these and the functions yT(ts, t)
ax:
Q.H. a
s=t8n L. at
v=4? Ls (Y:?.T4q) Its4= .778
N sGs I aY;
t;,-FhS
?GT dV-E?G-ILS (:S
In
R ? ,1,1
VI 4, 0 = 11:( 1 )i ? W 1
N s a (Nj lo=4.??1( 1-47)R+ill
J=o
(where n=1, 2, ..., Ms-1);
El?, (1.4 (0, t)
y('O, t)= GI dt + xl (0 , t);
4, dx11 (1, . t)
yli'. (1, t)= ---? dt +4(1, t);
G. 17
yl (1, t) .=-- yf (0 , t) + .-.- y i F;
y: (1, t) =
Ga+1
Gs
I 1 \ i a jw , r
1_; d (S+1 )i -r ? (L. (1, 0
) -
Es dx1 (1, t) [(
R dt -I ?L N'' )R+1]
(for s=2, 3, ..., b ?1).
The integrals are approximated by quadrature formulas to obtain the final system of ordinary differential
equations for the approximate values of the concentrations x(t) in (2), while the multiplicative forms Vis.
are approximated by Lagrange interpolation formulas, and the functions 4(t5, t) are approximated by Her-
mite formulas with two repeated points [4], followed by solution for the time derivatives.
The final form for the system of ordinary differential equations is
(2)
b - 1
dx:n (t)
dt E 2 [4,1?(t)+BITy:n (01, (3)
S-=1 n=1
where i=1, 2, ..., m; x:n(0)?xg.
The constants AS and Bn in (3) depend on the step length, the type of quadrature and interpolation
formulas, and all of the stage parameters.
The above method gives well-defined matrices for the coefficients An and BVn, and so it is permissible
to use the simplest methods of numerical integration in solving (3). It has been found that production of
solutions correct to two significant figures requires the points to be taken on the average at intervals of
100 separation stages, with never less than three approximation points per section. The length of the time
integration step is about 0.1 day for Euler piecewise methods. The Adams and Runge ?Kutta methods are
not appropriate in this case.
As an example of this method, Fig.1 shows the results obtained for the separation of a four-component
mixture of sulfur isotopes in a two-section stage with the two sets pi and 132 with an infinite pool at the intake;
the following parameters were used;
xiF = 0.0076; x2F =0.0422; x3F -- 0.00014;
Ni =N2= 400; Q'=0.0027 mole?cm-1;
C22 0.0004 mole. cm-1;
H'=20 m; I/2=12 m; p1=0.028 mole/day;
p2=0,01335 mole/day; Mi.?n=0,01 mole;
Li= 38.88 mole/day; L2=7.776 mole/day; 814 = 0,006; e24=0.011; 634=0.022.
594
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The order of (3) is in this case 24; for a process lasting 100 days, the solution requires 3 min on an
M-220 computer, with a maximum error of 0.2%.
LITERATURE CITED
1. A.M.Rozen, Theory of Isotope Separation in Columns [in Russian], Atomizdat, Moscow (1960).
2. R. Ya.Kucherov and V. P.Minenko, At. Energ., 19, 360 (1965).
3. I.A.Vereninov et al., Kernenergie, 14, 190 (1971).
4. I.S.Berezin and N. P. Zhidkov, Computational Methods [in Russian], Vol.1, Fizmatgiz, Moscow (1959).
595
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Declassified and Approved For Release 2013/03/01 : CIA-RDP10-02196R000300100005-6
ASYMPTOTIC STABILITY OF A REACTOR WITH
THERMIONIC CONVERTER
V. P. Gorbunov, N.V. Isaev,
0. V. Komarov, A. V. Kryanev,
N. N. Petrov, A. F. Piskunkov,
and S. B. Shikhov
UDC 621.039.514
The problems of dynamics and stability of a reactor, in whose active zone thermionic converters con-
nected with the fuel elements are placed, still remain open in the general nonlinear form, although articles
have appeared which consider either the linearized model [1] or a model taking account of the effect of de-
layed neutrons and the construction of the converter in a simplified way, where it is possible to use Belton
criterion for such simplified cases [2].
We consider a reactor in which the active zone consists of electrogenerating elements described in
[3]. Keeping in mind the results of investigations discussed in [4], a reactor with thermionic converter can
be regarded as an object with lumped parameters:
dW (t)
a
i = 1, 6.
separate parts of the active zone:
4
6k= E -c,;17';(t)-T;(0)1,
(1)
(2)
(3)
dR Ot w
= R I (t),
dt Ti
The reactivity ok is a function of the temperatures of the
1111111! ilit ii till It iii
5 10 15 20 25
W(0), MW
Fig. 1. Variation of the coeffi-
cient B as a function of the re-
actor power W in idling regime
for different load resistances
(R1 = 6.72 ? 10-3 n; R5 = 1.34
? 10-3 si; R9 = 0.11 10-3 2).
where ? is the reactivity coefficient for temperature T. ? The heat
3
balance equations for the fuel, the cathode, the anode stack, and the
heat-transfer agent can be written in the form:
dr{
Mf f dt ? ?al (Tf ? Tc)?bW;
Mc Cc dr = (T1? Tc)?a2 (T c? T a)?creFc(T-71)
dt
To PC (xa+ 2keTae + v1) ;
J2 (Ta, Ta) F R (Tc,
M aC a dT a dt = a2(Tu ?Ta)-03 (Ta ? Th)
-?"Fc(TC 71) + 12 (Tc' 7' a) IP el'
(Tc Ta) (Xa 2k7:: ) Fc;
mhch =a3(T a ? Th)? (14 (T h? Th. ,u),
(4)
(5)
(6)
(7)
where W(t) is the power of the reactor; Ti is the mean temperature;
M? C? are respectively the mass and the specific heat of the j-th
3' 3
Translated from Atomnaya t'nergiya, Vol. 32, No.6, pp. 501-503, June, 1972. Original article sub-
mitted October 7, 1971; revision submitted January 10, 1972.
596
C 1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York,
N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without
permission of the publisher. A copy of this article is available from the publisher for $15.00.
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Declassified and Approved
10 15 20 25
Wo mW
Fig. 2. Admissible deviations of
cathode and anode temperatures
from their stationary values for
different load resistances (121
= 6.72 .10-3 f2; R5 = 1.34 ?10-3
R9 = 0.11 -10-3 0).
MC
c c
MaCa
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layer, Xa is the work function of the anode surface; Tc7 is the tempera-
ture of the electrons at the anode; k is Boltzman constant; e is elec-
tron charge; b is the fraction of energy-release in the fuel; j is the
density of current from the surface of the cathode; Rel is the effective
resistance of the electrode; V/ is the voltage drop in the load; ai are
positive constants proportional to heat-transfer coefficients or to ther-
mal resistance of the layer; a is Stephen?Boltzmann constant; e is the
effective degree of blackness; and Fe is the emitting surface of the
cathode.
Experimental volt?ampere characteristics of the thermionic
converters, obtained in isothermal regime [5], were used in the solu-
tion of the above equations of dynamics. The temperature dependence
of the current density was described by an empirically chosen law:
(Tc, Ta) fa (TaTc? , (8)
where is a positive nondimensional quantity which can be assumed
constant for a certain neighborhood near the stationary value of the
anode temperature; j0 is a constant characterizing the slope of the
experimental curves. It is convenient to use the dependence (8) in
the analysis of the stability of the stationary regime:
W = W (0); R R (0); T j = 7' j (0) T u ?
In Eqs. (1)-(7) we make a change of variables:
w
W(0)
ri (t)= R, (t)? (0)
ri(t)> --1;
Ili (0) '
(t) = T j (t)? T, (0) , x j (t) > ? T j (0),
after which they become
6 6
?
13i (r+1);
i=1i=1 ii
Ti; (W +1)? (ri _1 1); i= I , 6.
al bW (0)
zv;
xf? Mf Cf (xf'?xc Ai f Cf.) I
az h
MC (37e xa)? m c ii tic (0) I xc Ta (0) } xal -- i[re (0), Ta (0)11
c c c c c c
{[Tc (0)+xe4 ?[Ta (0)-1- xayl- T6 (0)+21(0)1+ fr:2Ca {j2 (T (0) I Xe , Ta (0)-k:cal? /2 re (0), Ta (0)11;
? az aa
xa=- maca(xc xa)? maca aMCaTc(0)+xe Tand-xal?i Erc (0), Ta (0)11
.
{ Erc 1-. 47'
xc)4 a (0)4-xarl?T (0)+T: (0)1+ maca {}2 1Tc (0)-1-x0 Ta (0)-1- xal ? i2 [Tc (0), Ta (0)]};
a3
xh=i7th'h 3.111 M/Ci
where c = egFe, h = Fu(Xa + 2kT:), d2= FC(Rel +111), d3 = FUej., Tu = const.
The nonlinearities in Eqs. (16) and (17) are such that it is possible to separate out the nonlinear parts
in them. We write Eqs. (15)-(18) in vector form
ex bw (o) w+ z (re, xa), (19)
where 6 is a nonsingular matrix composed of the constants of the linear part of Eqs. (15)-(18); the vector
Z is composed of the nonlinear terms of these equations, while the vector b has the form
(b/M1
597
Declassified and Approved For Release 2013/03/01 : CIA-RDP10-02196R000300100005-6
Declassified and Approved For Release 2013/03/01 : CIA-RDP10-02196R000300100005-6
Let us assume that W(0), the steady-state power, belongs to a range of powers, where the matrix a
has different negative eigenvalues. We shall reduce matrix a to the diagonal form by the substitution X
= 13Y:
j bW (0) (13-1).i1 (p.--1)j2z2+ (P-1)i3Z3,
mf.Cf
whereX ? are the eigenvalues of matrix 6.
J
We write Lyapunov function of such a system of equations in the form
6 4
1
L =1[w? In (w 1).1 E (r?In (ri + 1)] +7 4111.
(20)
(21)
It is obvious that the function L is positive definite in the variables Yj, ri, w, if all Ai > 0. The total time
derivative of L, computed in accordance with Eqs. (13), (14), (20) has the form
6 4 4 4
dL
dt (w? r i)2 2 E (w +1) (71-1-1) Aix,y1+z, A.,(p-i)hyj+ Z3 4.1V. Ai (f)--1).133'1.
I
The quantity Aj is chosen in such a way that the condition
4
M1Cf E ce1 (i))11
1-1
Ai? >0
b W (0) (P-1)ii
is satisfied.
(22)
(23)
The condition for asymptotic stability is the requirement that the derivative dLidt be negative definite.
The first term in (22) is negative in view of the nonnegative nature of the variables w.+ 1, ri + 1; therefore
in the case of asymptotic stability the following inequality must be satisfied;
[6].
Thus it is possible to formulate a criterion for the sufficient conditions of the stability of the point
model of the reactor with thermionic converter. In order to ensure the asymptotic stability of the reactor
it is sufficient that all the eigenvalues of matrix C be negative, all the temperature coefficients of reactivity
4
satisfy the inequalities (1/(11-1)ii) cti(Ni > 0, while all the variables occurring in the nonlinear terms of
the feedback equations satisfy inequalities (24). Since the second method of Lyapunov gives only the suffi-
cient conditions of asymptotic stability, the actual region of stability may be much larger than that deter-
mined by the method indicated above. The dimensions of the region of stability are determined by the operat-
ing regime of the reactor. In the idling regime the current flowing through is equal to zero and the heat
transfer from the cathode to the anode is accomplished only by radiation.
The range of stability is defined by the inequalities
lelx4cf-f(xc)11