SOVIET ATOMIC ENERGY VOLUME 18, NUMBER 5

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• Collection: General CIA Records
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• Document Page Count: 138
• Document Creation Date: January 4, 2017
• Document Release Date: September 24, 2013
• Sequence Number: 5
• Publication Date: May 1, 1965
• Content Type: REPORT

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_
Volume 18, Number 5
May, _.1.65
SOVIET
ATOMIC
ENERGY
? ATOMHAfl 31-1EPITIR
.(ATOMNAYA iNERGIYA)
TRANSLATED FROM RUSSIAN
re
CONSULTANTS BUREAU
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NUCLEAR SCIENCE
MP iir-11. - ? wAkIr
W IFw
? ? ?ofoisw?? ??????-e-
? ???*** ???? ??? ? ??
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1?. 0.? ? - AA' iLw-
MOSSBAUER EFFECT METHODOLOGY,
Volume 1.
Erwin J. Gruverman, Editor
Proceedings of the First Symposium on Mossbazier Effect
Methodology, held in New York City on January 26, 1965,
The editor, a well-known and prolific writer in the Moss-
bauer field,' briefly reviews applications, then clearly and
accurately discusses the previously inaCcessible methodol-
ogy of Mossbau,er-effeet studies.
Because the book carefully examines the complexities
of required equipment for velocity modulation, measure-
ments of effects, and modification of external environ-
ments, the reader can find information not only for
evaluating the potential of Mossbauer-effect Studies, but
also for estimating the merits of alternative means of apply-
ing the iechniques. The comprehensive reviews by Herber
and SRijkerman describe new techniques for absolute
standardization of isomer shifts, using sodium nitroprus-
side standard absorbers. This will make possible adoption
of a standard to eliminate the now-Prevalent confusion in
comparing work performed at different laboratories.
Chemical-process industrialists, physicists, biologists,
chemists, and educators working in the Mbssbauer field ,
, should find this book 'of immeasurable help. ror example, '
it will aid readers in selecting a transducer system, build-
ing it, or writing specifications for its purchase; in assess-
ing the merits of the Mossbauer technique; and in con?
structing and specifying 'measurements, calibration and
environmental control systems. The, excellent coverage
of research and problems in this methodology includes
ninb carefully labeled diagrams of the apparatus and five
pertinent graphs, together with legend:
CONliNTS:. Mossbauer spectroscopy: some recent applicationS to
chernical.probleMs, Rolfe`H. Herber ? Review of advances in physics,
S. L. Ruby ? Application of the Mossbauer effect to biological sys-
tems, U. Gonser (and R/w.- Grant ? Feedback in electrochemiCal
drive systems, E. Kankeleit ? Cam driven, conitant acceleration
MOssbauer spectrometer, Alan I. Bearden, M. G. Hauser, and P. L.
Mattern ? Constant v6Iocity Mossbauer drive system. Flinn ?
Measurements by scattering techniques?I, I. K. Major ? Measure-
menta by.scattering techniques-4I, P. Debrunner ? f measurement
with black,absorbers, I. G. Dash ? Standardization Of the differential
chemicalshift for Fie", ii. Spiikerman, F. C. Ruegg, and I. R. DeVoe ?
Computation of Mossbauer spectra, J. R. Gabriel ? Superconduct-
ing magnets ? applications to the Meissbauer effect, Paul P. Craig ?
MOssbauer techniques using. high field water-cooled solenoids.
Norman A. Blum ?-LoW temperature cryostats for Mossbauer exPeri-
ments, Michael Kalvius ? Nigh pressure techniques, R. Ingalls.',
2/0d pages PP 1965 $12.50
THEORY AND METHODS OF
NUCLEAR REACTOR CALCULATIONS
G. L Marchuk, Editor
Six of the 18 papers in the collection are concerned with ?
transport theory. The first two papers .(by Marchuk et
a/.) deal with the properties and application of even-order
Pr, approximations, applying the difference 'equation fac-
torization method. These papers consider the'distribution
of neutrons from a point source;, it is shown that the
techniques of renormalization, extrapolation and ,over-
relaxation applied to Sr, equations accelerate pointwise
convergence. Three fundamental papers on resonance
? absorption are concerned with improving the treatment of
deviations from the basic narrow resonance formulas, in-,
, eluding computation of the departure of the collision
density, from the narrow-resonance shape. The paper
(Shikhov and Abagyan) on a method for constricting
Multigroup constahts. in the resonance region, taking
heterogeneous effects into account, has fundamental in-
'terest to engineering physicists. Three papers report orig-
inal work on fast neutron cross sections, and six are con-
cerned with reactor calculation methods. The 'papers in
the collection employ unified methods of analysis based
) ? on a strict mathematical approach, with Subsequent ap-
plication of the results to concrete practical solutions. A
Special Research Report translated from Russian.
208 pages CB 1964 $,40,00
EFFECTS OF RADIATION
ON ,SEMICONDUCTORS
By V. S. Vavilov '
Devotld to the effects of electromagnetic and corpuscular
radiations on semiconductors, this new volume deals with
the processes of absorption of electromagnetic radiation,
photoionization and ionization by charged high-eneigy
particles and the principal types of recombination proc-
esses by which an excited crystal returns to its original
equilibrium state. Translated from Russian. ,
CONTENTS : Absorption of light by semiconductors ? Photoioniza-
flan and photoconductivity in semiconductors ? Ionization of
semiconductorS, by charged high-energy particles ? Radiative re-
combination In semiconductors; posiibility of the amplification and
generation of light usihg semiconductors' ? Changes in the proper-
ties of semiconductors due to bombardment. with fast electrons,
gamma rays: neutrons, and heavy charged particles ? literliture
cited ? Index.
238 pages B. 1965 : $15.00
CONSULTANTS BUREAU! PLENUM ,PRESS
227 *est 17th Street, New York, New York 10011
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Declassified and Approved For Release 2013/09/24: CIA-RDP10-02196R000700010005-2
ATOMNAYA NERGIYA
EDITORIAL BOARD
A. I. Alikhanov M. G. Meshcheryakov
A. A. Bochvar M. D. Millionshchikov
N. A. Dollezhal' (Editor-in-Chief)
V. S. Fursov P. N. Palei
I. N. Golovin V. B. Shevchenko
V. F. Kalinin D. L. Simonenko
N. A. Kolokol'tsov V. I. Smirnov
(Assistant Editor) A. P. Vinogradov
A. K. Krasin N. A. Vlasov
A. I. Leipunskii (Assistant Editor)
V. V. Matveev
SOVIET ATOMIC
ENERGY
A translation of ATOMNAYA ENERGIYA,
a publication of the Academy of Sciences of the USSR
0 1966 CONSULTANTS BUREAU, A DIVISION OF PLENUM PUBLISHING
CORPORATION, 227 West 17th Street, New York, N. Y. 10011
Volume 18, Number 5 May, 1965
CONTENTS
Equilibrium of a Spatial Plasma Pinch in a Longitudinal Magnetic Field Under Steady-State
RUSS.
PAGE PAGE
Conditions?V. D. Shafranov
575
443
Properties of a Thermionic Diode Placed in an Autonomous-Discharge Plasma
?N. D. Morgulis and Yu. P. Korchevoi
580
447
Coherent Effects During the Interaction of Slow Neutrons with Liquids?V. P. Vertebnyi,
I. P. Dzyub, A. N. Maistrenko, and M. V. Pasechnik
585
452
Total and Differential Fission Cross Sections of Uranium and Thorium for Low Energy
Deuterons?Yu. A. Nemilov, V. V. Pavlov, Yu. A. Selitskii, S. M. Solov'ev,
and V. P. gismont
590
456
An Exact General Solution in Spherical Harmonics of the Boltzmann Equation
?G. Ya. Rumyantsev
594
459
Calculation of Weak Self-Oscillatory Conditions in Nuclear Reactors?B. Z. Torlin
600
463
Propagation of Neutrons in Iron?V. I. Golubev, A. V. Zvonarev, M. N. Nikolaev,
M. Yu. Orlov, V. V. Penenko, and 0. P. Uznadze
608
469
Investigation of Power Effects of the BR-5 Reactor. N. V. Krasnoyarov, R. V. Nikol'skii,
and I. A. Efimov
613
474
Development of a Fabrication Technology for Organic-Coolant Purifying Filters,
and the Study of Their Hydraulic Resistance?Yu. I. Tokarev, F. F. Bogdanov,
E. L Pavlovskaya, and A. P. Chernopyatova
617
478
Internal Stresses Caused by Non-Uniform Swelling of Fissionable Material
?Yu. I. Likhachev, V. P. Zvonarev, and V. Ya. Pupko
623
483
Extraction of Radium from Liquid Waste by Sorption on Manganese Dioxide
?A. P. Tyutrina, B. P. Zhagin, and V. G. Bakhurov
628
487
Reaction Kinetics and the Equilibrium State in the System CO2?CO?C Under the Action
of Fast Electrons?G. P. Zhitneva, S. Ya. Pshezhetskii, N. A. Slavinskaya,
and S. A. Kamenetskaya
632
492
Sr9? Content of Radioactive Fallout in Western Slovakia?Sh. Chupka, M. Petrashova,
and I. Tsarakh
637
496
Use of the Autoradiographic Technique for Studying Radioactive Aerosols ?V. N. Lavrenchik .
640
499
Pb21? in the Atmosphere and in Fallout?V. I. Baranov and V. D. Vilenskii
645
503
ABSTRACTS OF DEPOSITED ARTICLES
Investigation of the Phonon Spectrum in the Copper Lattice by Using the Method of Inelastic
Neutron Scattering?E. Z. Vintaikin, V. V. Gorbachev, and P. L. Gruzin
649
507
Annual Subscription: $95
Single Issue: $30 Single Article: $15
All rights reserved. No article contained herein may be reproduced for any purpose whatsoever
without permission of the publisher. Permission may be obtained from Consultants Bureau, A
Division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011, U.S.A.
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Declassified and Approved For Release 2013/09/24: CIA-RDP10-02196R000700010005-2
CONTENTS (continued)
RUSS.
PAGE PAGE
Turbulent Couette Flow?V. D. Vilenskii and V. P. Smirnov
650
508
REVIEWS OF GENEVA 1964 PAPERS
Nuclear Power Plants for Civilian Maritime Use?N. S. Khlopkin
653
510
Science and Engineering Exhibits at the Third Geneva Conference ?B. A. Kuvshinnikov
and V. V. Frolov
656
511
LETTERS TO THE EDITOR
Multibeam Radio-Interferometer Determines Plasma Parameters?V. Ya. Balakhanov,
V. D. Rusanov, and A. R. Striganov
660
515
Bremsstrahlung and Characteristic Radiation Spectra of Zirconium-Tritium Sources
?Yu. P. Betin
663
516
Determination of the Relative Fission Reaction Rates for Different Isotopes by Recording
the y-Radiation of Lal" Fission Fragments?L. N. Yurova and A. V. Bushuev
665
518
Energy Distribution of ot Particles Emerging from a Thick Source?p. B. Ershov,
A. A. Karan, and V. P. Shamov
667
519
Parabolic Approximation of the Total Attenuation Coefficients of y-Quanta in the Energy
Range from 0.03 to 10 MeV-0. S. Marenkov and R. S. Derzhimanov
669
520
The Effect of Boron-Containing Blocking on the Yield of Capture y-Radiation
?S. A. Kozlovskii, V. S. Kyztyurov, K. K. Popkov, and D. N. Lebedev
672
522
Pulsations of the Pipe Wall Temperature Under Conditions of Intensive Convective
Heat Exchange?V. I. Subbotin, M. Kh. Ibragimov, V. I. Merkulov, E. V. Nomofilov,
and N. A. Tychinskii
676
525
Experimental Investigation of the Thermal Conditions of Fuel Elements in the VVR-M
Reactor?I. F. Barchuk,.M. M. Nazaxchuk, S. S. Ogorodnik, D. T. Pilipets,
and S. 0. Slesarevskii
679
528
Attenuation by Iron and Polyethylene of Tissue Dose of Neutrons Incident Obliquely
on the Shielding?G. V. Miroshnikov
681
529
Attenuation of Neutron Tissue Dose by Thin Layers of Hydrogenous Materials
?G. V. Miroshnikov
685
532
Fractionation of Radioactive Isotopes in Hot Particles?Ya. I. Gaziev, S. G. Malakhov,
and L. E. Nazarov
689
535
SCIENCE AND ENGINEERING NEWS
Dubna August 1964 International Conference on High Energy Physics
692
538
II Colloquium on Inelastic Scattering of Slow Neutrons in Crystals and Liquids
?V. V. Golikov
699
543
NEWS
The Beta-2: New Isotope Electric Power Source?G. M. Fradkin, V. M. Kodyukov,
and A. I. Rogozinskii
702
545
General-Purpose Gamma-Ray Device Designed for Pilot-Plant Radiation-Chemical
Processes with Displacement of Radioactive Co" Preparations by Compressed Air
?V. I. Volgin, V. E. Drozdov, M. E. Eroshov, G. L Lisov, A. N. Neprokin,
and Yu. S. Ryabukhin
704
546
The Russian press date (podpisano k pechati) of this issue was 5/8/ 1965.
Publication therefore did not occur prior to this date, but must be, assumed
to have taken place reasonably soon thereafter.
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EQUILIBRIUM OF A SPATIAL PLASMA PINCH
IN A LONGITUDINAL MAGNETIC FIELD
UNDER STEADY-STATE CONDITIONS
(UDC 533.9.07)
V. D. Shafranov
Translated from Atomnaya Energiya, Vol. 18, No. 5,
pp. 443-446, May, 1965
Original article submitted September 5, 1964
The equilibrium conditions of plasma in toroidal chambers of the figure-of-eight type are determined
for the case in which the plasma is preserved for longer than the magnetic field takes to pass through
the windings of the solenoid. In this case, some of the magnetic lines of force pass through the solen-
oid windings, closing outside it. Hence, the radius of the cross section of the extreme toroidal mag-
netic surface lying as a whole inside the chamber is always smaller than the radius of cross section
of the solenoid. For a plasma pressure exceeding a certain critical value, toroidal magnetic surfaces
are absent, and containment of the plasma is in principle impossible.
The simplest system for containing a dense plasma in an external magnetic field created earlier is a closed
solenoid having the form of a space curve (for example, a figure of eight). This sytems was proposed by Spitzer [1],
who also made a qualitative estimate of the equilibrium conditions of plasma in a solenoid in the shape of a figure
eight. A more precise derivation of the equilibrium conditions, based on a calculation of the equilibrium position
of a plasma pinch inside a solenoid of given form, was made in [2, 3]. In these papers, however, it was assumed
that the surface of the solenoid (chamber) was an ideal conductor and, hence, coincided with the extreme toroidal
magnetic surface of the equilibrium configuration. In this case equilibrium is in principle possible for any values
of the parameter 3 = 87rp/B2; this follows, for example, from [4]. In practice, however, the severe distortion of
the magnetic surfaces arising for relatively large values of 13 is undesirable. From the requirement that the distor-
tion of the magnetic surfaces should be small we may define an arbitrary critical value 5 cr for plasma equilibrium
in the ideal solenoid. This arbitrary criterion was also determined in [2, 3].
The condition that a solenoid made of ordinary (nonsuperconducting) materials should be ideal is only satis-
fied for periods small compared with the time of penetration of the magnetic field through the walls of the solenoid,
which in experiment is some hundredth parts of' a second. Hence, if the plasma containment time is of the order of,
or greater than, 10 msec (such orders of containment time are already met with in experiments), the equilibrium
conditions obtained on the assumption that the chamber is ideally conducting become inoperative.
The aim of this paper is to find the equilibrium conditions for a fairly large containment time, when the con-
figuration of the magnetic field is determined not by the position of the conducting surfaces but by the assigned cur-
rent distribution on these surfaces (steady-state conditions). This is formally expressed by the fact that the condition
that the normal component of magnetic field on the surface of the solenoid should vanish is replaced by a condition
expressing the difference between the tangential components of the magnetic field in terms of the given value of
surface current density in the solenoid (the thickness of the solenoid windings being neglected), and the condition
that the normal component of magnetic field should be continuous.
If the coils of the solenoid are disposed in planes perpendicular to its axis, then, as we know, for a toroidal
axially-symmetric solenoid, all the magnetic lines of force pass inside the solenoid. In a solenoid having its axis
in the form of a space curve, however, under steady-state conditions some of the lines of force penetrate the surface
and close outside. The process of establishing a magnetic field in such a solenoid takes place as follows. When the
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current is rapidly switched on, a magnetic field is set up in the solenoid with the same configuration as if the sole-
noid had ideal conductivity. Then the magnetic field partly diffuses through the windings of the solenoid, so that
the outer toroidal magnetic surfaces in the solenoid are disrupted. If plasma is placed in such a magnetic field,
then, as plasma pressure increases, the magnetic surfaces are not pressed to the walls of the chamber, as would be
the case for ideal conductivity, but unimpededly pass through the surface of the chamber (solenoid). For a fairly
large plasma pressure the toroidal magnetic surfaces vanish completely, when the magnetic axis shifts to the surface
of the solenoid. Thus, in steady-state conditions, in contrast to the case of an ideally-conducting chamber, where
equilibrium is possible for any plasma pressure, there is a certain critical value 6 a above which equilibrium is in
principle impossible (magnetic toroidal surfaces absent).
A method of calculating plasma equilibria in spatial solenoids is described in [2, 3]. This is based on the use
of a quasi-cylindrical orthogonal system of coordinates Q, s, co, [5] in which Q is distance reckoned from the axis of
the solenoid, s is the axis arc length, and w is the azimuthal angle reckoned on the surface Q = const from a line
perpendicular to the sections s = const. The toroidal corrections to all the quantities are expanded in Fourier series:
co
i(co?xos)-ki ns
B(1) = Re E 137, ((I) e
The equation of a magnetic surface with cross-sectional radius Q' has the form
(1)
(2)
, 71S
= + Re (Q )e
77= -00
where xo is the mean value of the rotation angle a(s) of the principal normal to the axis of the solenoid, relative to
the line co = const.
a ([4
xo ? L
(3)
where L is the total length of the axis.
An equation for determining g n(Q) was obtained in [3], together with expressions for the Fourier components
of the toroidal corrections to the magnetic field, plasma pressure, and plasma current density in terms of gn(Q) and
dgn/dQ. All these formulas appear simplest when the curvature k(s) and the rotation angle of the principal normal
a (s) are smooth functions of arc s. Let us consider just this case. For a figure-of-eight consisting of circular arcs,
the results given below are valid everywhere except for a circle of points where the arcs join over a length roughly
equal to the diameter of the solenoid (As P..42b). In the approximation indicated, the equation for n(Q), in the ab-
sence of longitudinal current, is obtained from Eq. (102) of [3] in the form
,rt (Q) = kn 8TE [(P)p?--Po (Q)1 3
(1Q x B0 4 71Q*
(4)
Here p is the pressure averaged over the cross section of radius Q; the quantity -x,n is connected with xo [see
Eq. (3)] by the relation
25t
xn = xo n?
(5)
The condition (mentioned above) that the functions k(s) and a(S) should be smooth corresponds to the limitation
x?cr < 1. (6)
The Fourier components of the toroidal corrections to the magnetic field, current density, and plasma pressure
are expressed by the formulas:
576
? xan,Bso; (7)
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Bwn -= 42+'n (+Q d): Bso:
4n . ?
.13,ft knQB so;
1. paIwo,
Jwn?(1 dt? +knQ)
kn .
isn ? lwo;
xn
dpn
Pn d6- ?
Here kn is the Fourier component of the relative curvature K(s)
; 2n
kn=- K (s)e-- L vs ds;
K (s) = k (s) ei[xos?a(s)],
which is easily calculated from the known functions k(s) and a(s) [3].
Let us denote by a the radius of the plasma pinch, i.e., the radius of the cross section of some magnetic sur-
face on which the current density and pressure gradient are negligibly small. Then for Q a- a the mean pressure in
formula (4) is
a2
(P)a (16)
the displacement of the plasma pinch of radius a in an ideally-conducting solenoid of radius b, determined by inte-
grating expression (4) from b to a, takes the form [3]:
3
where
(17)
87t (p), (18)
Blo ?
In the case of our poorly-conducting solenoid, however, where the extreme toroidal magnetic surface has a cross-
sectional radius smaller than that of the solenoid, in order- to determine the equilibrium position of the plasma pinch
we must solve the problem of the magnetic field under the following conditions:
1) on the surface of the plasma pinch, in accordance with the definition of this surface, the normal component
of magnetic field vanishes;
2) the tangential components of magnetic field on the surface of the plasma pinch are determined by the
values (8) and (9) obtained from solution of the internal problem;
3) on the surface of the solenoid the normal component of magnetic field is continuous;
4) the jump in the tangential component is determined by the given distribution of currents on the surface of
the solenoid.
577
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Since the solution procedure is extremely simple, we shall confine ourselves to giving the results obtained.
The expression for the magnetic-field components between the pinch and the solenoid may be taken from [2]:
B pn i n Bso+ AI; (xn.Q) ? BniCi(xn9);
kn
Bton Bs0 + [Anil (HO) B (x 0Q)];
xn xnQ
B8 = ? i [417. I (xnQ) BnK (xn,Q)1;
where II and K1 are Bessel functions of imaginary argument. Outside the solenoid:
B pn = C ?K; (xnQ);
B.= CnKt (xnQ); (22)
= ? iC (xnQ) ?
The constants An and Bn are obtained from the first two conditions with the allowance for relations (4), (8), and (9),
taken for Q = a, in the form:
kn ,
A0= bda2B so 71?51t n2 kx?,a)
? x?,
81((p)n? ?3 x2 a2B,io
2 " (23)
iknx?a2K; (xna)
Bso
3
4. 82-t (p)a ---2 sq,a2M0
B? = ix;,a2Bso n ?71) /2 (xna)? ik?x?a2/; (xna) (24)
so
Since relations (4), (8), and (9) were obtained on the assumption that xnQ ? 1, for determining An and Bn we must
also use the expansion of the modified Bessel functions in terms of xna. From the third and fourth conditions we
find Cn and En. Let us suppose that, in addition to the current created by the main magnetic field, the surface of
the solenoid also bears a longitudinal current of the dipole type, by means of which the position of the magnetic
surfaces can be regulated:
2n
i(o)--xns)+i ?ns
=Re ime L
71-= ? Oo
Then the expression for Cn takes the form
k?X3nb4
k?X?a2 0
2Jt ; 2 A )
C0 ? 8 Bso 2 p,uso c snxn
and the unknown Fourier component of displacement is
k' b2 ( 3 02
'4 1 2 b2
2at b
' e x0 Bo ?
A characteristic feature of equilibrium in the conditions of a poorly-conducting solenoid considered is that the dis-
placement due to the Plasma pressure (second term in the expression for En) is independent of the radius of the plas-
ma pinch. For a certain Critical value of parameter 13 the displacement becomes comparable with the cross-sec-
tional radius of the solenoid. Hereupon, the cross-sectional radius of the extreme toroidal magnetic surface touching
the walls of the solenoid vanishes, and equilibrium becomes impossible in principle.. By comparing expressions (27)
and (17) we see that the displacement Connected with the pressure in an ideally-conducting sheath differs from that
in a poorly?-coriduCting Solenoid by the coefficient 1,,.a2/b2. This means that there is no need to make a numerical
Sn
kn. p,
(25)
(26)
(27)
578
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calculation for the displacement g (a, co, s) in a poorly-conducting solenoid, since we may use the results for the
ideal sheath given for certain configurations in [3].
Another characteristic feature of equilibrium in a poorly-conducting solenoid is the possibility in principle of
experimentally determining the energy of the plasma from measurement of the magnetic field outside the solenoid.
This field (dipole type) is determined by the formulas
Cn Cn
B = Bum
pn 0, x22
IL
i.e., according to Eq. (1),
Bco(Q, (0, s) b2
B20 Q2
B20
k12xnb2 a a2 kn
8 P b2 2x?
- 2a
cos (o) X08 + ns ;
k12x?b2 a a2 kn
8 b2 2 sinx?
27c
? xos +? ns .
}
(28)
(29)
(30)
In order of magnitude k0/x43 s?11. Hence, this dipole field close to the surface of the solenoid is approximately a
fraction 8 of the main field in the solenoid and should be easy to measure. Moreover from formula (29) or (30) we
may find the value of 8a2, proportional to the thermal energy of the plasma. A detailed calculation of the external
field [associated with the summation of the series in Eqs. (29) and (30)] can easily be made in any specific case by
analogy with the calculation of displacement made in [3].
In conclusion we note that, when the plasma contains a longitudinal current J and an associated field 13?0(Q)
= 2J/cQ, the formula for the displacement is more complicated. For reference we give this without derivation
2n b isa k1252 3 a2
xn Bso 4 2 b2
k B1 (a) kna[ 8It (p) a + Sc). On the other hand, for an all-around increase in the diode current, we must have a cathode with
a large emitting power and also ensure that the electron space charge near the cathode is effectively neutralized
[5]. For this, we need a corresponding concentration of charges in the autonomous plasma, since the flux of electrons
from the cathode does not normally give any marked additional ionization. For this kind of neutralization we must
clearly satisfy the condition
=0.4en 2kTe p v m Jc V7/17
kT
= en e 2arne 0,74 ,
(4)
which characterizes the degree of ionization of the plasma necessary for the operation of the diode. Here we must
remember the effect of Je and Jp on 6 and V [formulas (1) and (2)].
We thus see how important it is to make a systematic experimental investigation of the properties of such di-
odes as a physical model of the "thermionic-plasma? energy converter. In the first stage of our work on this subject
(reported in [1, 2]), we studied the properties of a diode placed in an autonomous discharge plasma in cesium vapor.
This plasma is of special interest in connection with its wide use mainly in the ordinary kind of thermionic energy
converter. It should be noted that the parameters of the cesium plasma obtained in the latter case are determined
by the operating conditions of the diode system as a whole, and, hence, it is practically impossible to control them
independently. Moreover, the electron temperature for operating conditions typical of these converters is extremely
low (order of 3000?K) [6]. Hence, the output voltage of the ordinary type of converter is insignificant, even for
AVc 0 (order of 1 to 2 V).
In our paper on the diode in an autonomous cesium discharge plasma [1, 2] for AVc 0, we gave some experi-
mental data which enabled us to estimate the degree of accuracy of the considerations underlying formulas (1) and(2).
Figure 1 (curve I) shows the relationships & = f (ne, Te) as obtained experimentally and by calculation from
formula (1). As we see from Fig. 1, these curves are in close agreement. In the same figure (curve II) we have the
relationship 6 =f (log Jc) plotted from experimental data; in accordance with formula (1), in this case we obtain a
straight line, the slope of which gives the value of Te ? 3000?K, which agrees closely with the true value.
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Fig. 3. Volt/ampere characteristics of the current
of a plane-parallel diode in mercury plasma for
the cases: Te =.30,000?K; I) ne = 8 ?109
-3.
C111 II) Re = 4.1010 cm-3; III) ne = 4.1011 cm-3;
Te = 15,000?K; I) ne = 3 ? i0 cm3;
-3
II) Re = 2 ? 1010 cm; III) ne = 1.1011 CIT1-3.
I
/
/
/
/
/
/
/
/
/
/
V,V -12
-8
J,A/cm2
? 0,3
0,2
0,1
0
Fig. 4. Volt/ampere characteristics of the current
of a plane-parallel diode in mercury plasma for the
case ne = 4.1011 cm-3 and Te - 30,000?K: --
-
experiment; calculated from for-
mula (2).
Figure 2 shows the experimental volt/ampere charac-
teristics on a semilogarithmic scale for various values of ne,
Te, T. Under these conditions the emf (represented in the
figure by the intercept on the axis of abscissa) may reach
considerable values (? 7 V). Comparison shows satisfactory
agreement between the experimental and computed data
obtained for various values of Te and Te, i.e., Jc.
In the second stage of the work it seemed natural to
study plasma in which the electron temperature was much
greater than that in the cesium plasma. This holds for a
discharge plasma in inert gases or mercury vapor [3]. We
found it convenient to use the plasma of a mercury-vapor
arc discharge, which had a fairly high Te, easily regulated
by the mercury-vapor pressure. Diode systems of two types
were introduced into this kind of plasma: 1) a plane-paral-
lel system (Sa = Sc) consisting of a small barium-impreg-
nated disk cathode, a disk anode, and a stationary probe set
alongside them in the plasma; 2) a cylindrical system con-
sisting of an ordinary heated oxide cathode, a wide corru-
gated anode (Sa ? Se) and a stationary probe introduced
inside this system. In all cases AVe = 0, since the anode
was doubtless covered with an active layer of evaporation
products from the cathode.
Figure 3 presents volt/ampere characteristics obtained
with the plane-parallel diode for various values of ne and
electron temperature 30,000 and 15,000?K. We see that as
Te rises there is a considerable increase in the emf and the
optimum output voltage of the diode Vm. On raising the
value of ne, i.e., Je and Jp, the short-circuit current J0 rises
and 6 falls. All this agrees with formulas (1) and (2). In
order to make this comparison perfectly clear, the experi-
mental and theoretical volt/ampere characteristics for cer-
tain values of ne and Te are shown in Fig. 4. The two curves,
reduced to the same value of short-circuit current, agree
closely with one another. Finally, we note that the experi-
mental curve gives extremely high values of & and Vm
(? 14 V and ? 5 V, respectively).
A comparison between the volt/ampere characteristics
for the plane-parallel and cylindrical diodes for the same
plasma parameters is shown in Fig. 5 (curves I and II). On
passing from the first system to the second there is a certain
increase in & and Vm. It is interesting to note that, when
the valUe of Te for the cylindrical diode was raised from
30,000 to 50,000?K for the same value of ne, curve III was
obtained; this gives still larger values of 6 and Vm (? 20 and ? 8 V, respectively), as would be expected. In com-
parison with these, the emf of the ordinary diode &' + 2k/e (Te?Ta)] is insignificant.
After elucidating these features, we passed on to a more detailed consideration of the phenomena in the inter-
electrode space of the diode in the mercury discharge plasma. For this we used an apparatus in which the diode sys-
tem to be studied was situated at floating potential in the mercury-arc discharge plasma (apparatus shown schemat-
ically in Fig. 6). The cathode of the diode was a chain of seven heated oxide cathodes in parallel on the surface of
a 5-mm diameter cylinder; the anode was an ordinary molybdenum cylinder 7 mm in diameter and 18 mm in length.
On both sides of the diode (close to it) lay two stationary probes, enabling the potential drop AVg in the plasma in
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Fig. 5. Volt/ampere characteristics of the current of
a diode in mercury plasma for the case* ne=1010 cm-3;
Te = 30,000?K and short-circuit current jo = 0.1A/CM2:
1) plane-parallel diode; II) corrugated cylindrical di-
ode; III) the same cylindrical diode at Te = 50,000?K.
X
Cathode
Anode
V,V
30
20
Tie, CM-1
3 401?
2.101?
1401?
A
.4"
10
0
-10
-20
0 2 4
6 ;mm
Fig. 7. Distribution of potential V and electron concen-
tration ne inside the diode for discharge current 0.1 A
in the plasma and Te ? 80,000?K for the emf (
and short-circuit (---) conditions.
Fig. 6. Experimental apparatus for studying mer-
cury-plasma parameters inside the cylindrical di-
ode: 1) internal movable probe; 2) two external
stationary probes; 3) guiding glass tube; 4) cath-
ode; 5) anode of the diode.
CathOde
71e, cm
2.1011
1-10"
Anode
V, V
20
I?
_
0 2 4 6 z , mm
Fig. 8. Distribution of potential V and electron concen-
tration ne inside the diode for discharge current 1.0 A
in the plasma and Te ? 50,000?K for the emf (
and short-circuit (---) conditions.
this section to be determined. Inside the diode system, a third probe was introduced through a small hole in the an-
ode; this could be moved over the radius. The probe in the interelectrode (cathode?anode) space was used to de-
termine the distribution of potential V and the temperature Te and concentration ne of the plasma electrons. Meas-
urements of this kind were made for the erhf and short-circuit (ZSVe = 0) conditions of the diode.
Figure 7 gives the distributions of potential and electron concentration obtained for the case in which the dis-
charge current in the plasma i = 0.1 A; here, it was found that Te ? 80,000?K = const, and LVg = 8 V. On the other
hand, measurement of the volt/ampere characteristics showed that in this case e = 17 V and the optimum output
voltage Vm= 7 V. Figure 8 shows results obtained for the case in which the discharge current in the plasma i = 1.0 A.
For this Te ? 50,000?K = const, AVg = 5 V, e = 12 V and vm = 5 V.
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From the data presented in these figures we may draw the following conclusions.
1. The distribution of potential in the interelectrode space of the diode V = f(x) shows that, first, in accord-
ance with formulas (1) and (2), the large values of 6 and Vm (agreeing with Te) are due mainly to the large nega-
tive potential jump Va at the anode [4] (for example, in these experiments Va reached 30 to 50 V in the emf con-
dition); secondly, within the diode there is a considerable and smooth rise in potential, and, hence, a substantial
cathode potential jump; as a result of the latter, electrons coming straight from the cathode may play some part
in the formation of charge in the plasma.
2. The electron concentration tie measured inside the diode is considerably smaller, and the electron temper-
ature Te and longitudinal electric field E = LVg/d considerably larger, than under the same conditions in "free"
plasma (according to the results of [3] ,and of the measurements indicated above). This is connected with the addi-
tional increased recombination of plasma charges at the electrodes of the diode, similar to the well-known condi-
tions in a plasma pinch.
Finally, we should mention some further experiments confirming that a slight longitudinal and transverse mag-
netic field may have a considerable effect on the parameters of the diode shown in Fig. 6. For example, for a dis-
charge current of i = 0.1 A in_the plasma and a magnetic field of ? 150 Oe, the value of Vm reaches ? 10 V for the
longitudinal and ? 16 V for the transverse field, while the value of & in both cases is ? 30 V.
The results of our experiments indicate the interesting physical properties of such diodes and also show that
increased emf and output voltages can be obtained and controlled with them. We shall continue to study this problem.
In conclusion we must make two comments.
1. This kind of diode, as a physical model of a "thermionic-plasma" energy converter, can have nothing in
common with a converter using an auxiliary discharge as a source of the ions required for increasing the electron
current [7]. On the other hand, certain features of this diode are probably similar to those of the well-known con-
verter using an ionized-gas flow without a magnetic field [8]. The operating principle of our diode to a certain ex-
tent reflects the recent widespread tendency to use nonequilibrium plasma systems with "hot" electrons, as, for ex-
ample, in questions of the conductivity and electron emission of semiconductors, the conductivity of magnetohydro-
dynamic energy converters, and so forth.
2. After all our cesium-plasma work had been completed and [1, 2] had been published, a brief, independent
article [9] appeared. This contained, arguments similar to ours(though without experimental foundation) and also the
results of certain experiments directly demonstrating the genuine possibility of converting energy by means of this
kind of diode. Values showing the practical applicability of this method were obtained for an argon plasma created
by an hf discharge :& 10 V, Vm -- 5.3 V, and (according to the estimate of [9]) a conversion efficiency of ? 36%.
LITERATURE CITED
1. N. Morgulis and Yu. Korchevoi, ZhTF, 32, 1487 (1962).
2. N. Morgulis, "Ukr. fiz. zh.," 7, 1131 (1962).
3. B. Klyarfel'd, in the coll.: "Electron and Ion Apparatus" [in Russian], Moscow-Leningrad, Gosenergoizdat
(1940), p. 165; J. Phys., 5, 155 (1941).
4. B. Klyarfel'd and N. Neretina, ZhTF, 28, 296 (1958); 29, 15 (1959); 30, 186 (1960).
5. P. Marchuk, "Trudy IF AN USSR," No. 7, 17 (1956).
6. N. Morgulis and Yu. Korchevoi, "Radiotekhnika i elektronika," 6, 2073 (1961).
7. D. Gabor, Nature, 189, 868 (1961); W. Bloss, Z. angew. Phys.., 14, 1 (1962).
8. S. Klein, Proceedings of Fifth International Conference Ioniz. Phenomena in Gases, 5, 1 (1962), p. 806.
9. J. Waymouth, J. Instn. Electr. Engrs., 8, 380 (1962).
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COHERENT EFFECTS DURING THE INTERACTION
OF SLOW NEUTRONS WITH LIQUIDS
(UDC 539.17.02)
V. P. Vertebnyi, I. P. Dzyub, A. N. Maistrenko,
and M. V. Pasechnik
Translated from Atomnaya Energiya, Vol. 18, No. 5,
pp. 452-455, May, 1965
Original article submitted May 18, 1964
The total cross sections of liquid nitrogen and oxygen and the total cross section of gaseous nitrogen
for neutrons with wavelengths in the 4 to 15-A range were determined in the VVR-M reactor of the
Institute of Physics, Academy of Sciences, Ukr.SSR. The cross sections of gaseous oxygen and nitro-
gen rise monotonically with increasing wavelength, while those of the liquids begin to fall at
X 5-5.5 A. Apparently this effect is connected with the existence of short-range order in the li-
quids; in nature, it is similar to the scattering of neutrons in polycrystalline material for wavelengths
close to X = 2dmax. The existence of coherent effects in the total cross section of heavy water is
considered.
The total cross section for the interaction of neutrons with a liquid may in general be written in the form
G tot Gscatd G react. (1)
The question of the energy dependence of ?react is practically solved. The energy dependence of ?scat may be
found if we know the correlation function G(r, t) [1], since
d2ascat_ a2k
d52 de 2Trii k0 exp [i (kr? Gan G (7', t) dr dt.
In reality, however, the function GT, t) is unknown. If we put the total scattering cross section in the form
coh incoh
ascat=6elas+ G el as -1 inelas.
(2)
(3)
and suppose that the term oeeh is the greatest (this assumption is apparently valid for liquids at low temperatures
elas
with a large coherent cross section, for example, for liquid oxygen and nitrogen), then in the static approximation
(without allowing for loss of energy in scattering) we may obtain some information on the variation of oelas with
coh
neutron wavelength. In fact
cID
lidac.h._= a2c 4a@o [g (r)--- 1] sill sr
sr
df2
EE a2cohi'(s),
where acoh is the amplitude of coherent scattering, g(r) is the binary correlation function, Q0 the density of the
nuclei, s = 47T/X sin 0/2 (here 0 is the neutron-scattering angle). It follows from (4) that
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(4)
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2
6 7
r ?
A
;7,
000
A
Fig. 1. Arrangement of experiment to determine the total neutron
cross sections of nitrogen and oxygen: 1) reactor shielding; 2) col-
limator; 3) beryllium filter cooled with nitrogen; 4) mechanical
monochromator (or interrupter); 5) radiation shielding; 6) cryostat
for gas container; 7) Born counters SNMO-5.
50
3 4
75 100 125
Number of channel
I I
5 6 7 A,4
Fig. 2. Neutron spectrum measured by means
of a mechanical neutron interruptor with reso-
lution 16 psec/m, after passing through the
beryllium filter. The upper curve relates to
cooled beryllium.
coh.(k) dQ =
elas dS2 Jcoh
A 2 r 2
= --tnCtooh 4yt
4n
slim =
Si(s)ds.
(5)
Similar expressions may also be obtained in the approximation of
[2], where the quasi-elastic character of the elastic scattering is
taken into account. The quantity i (s), the 'Fourier transform of the
binary correlation function, may be obtained directly from x-ray
or neutron-diffraction data [3, 4]. The general character of i(s) is
this: for small s, i(s) is small; as s rises, the value of i(s) passes
through a sharp Maximurn connected with the existence of short-
range order in liquids; as s rises further, i(s) falls once more and
smoothly with s. Hence, for large neutron wavelengths the integral
47E
Slim
is small and so is the coherent-scattering
cross section. As the wavelength decreases, there comes a moment
when slim falls in the neighborhood of the maximum of i(s). Here
the cross section increases quite sharply. Finally, after passing the
i(s) maximum and continuing to shorter wavelengths, there is a
14ri
slight increase in the integral \ (s) s ds: the variation in the
factor X2 in front of the integral in (5) may be more substantial, so
that the coherent-scattering cross section will again diminish.
It is interesting to note that i(s) may be determined from measured values of the coherent-scatttering cross
section. In fact, if we introduce the notation t = 47r/x, it follows from (5) that
586
=
1 1 d 2 c oh
[ (a)].
42'ta2 cl; tot
cob
(6)
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1
0 4 8 12 16
Fig. 3. Neutron spectrum measured by means of a
mechanical monochromator after passing through
the beryllium filter: 1) immediate results of meas-
urements; 2) with the introduction of a cotrection
for attenuation of the neutrons in the air on the way
from the channel exit to the counter;, 3) with fur-
ther allowance for the attenuation of the neutrons
in the air channel of the reactor.
20
70
0
? O.
. ? 'a
o ? ? .0 0 o ?
2
5
10 A,A
Fig. 4. Variation of the total scattering cross section of li-
quid nitrogen with wavelength (measnred by means of the
mechanical neutron interruptor with resolution 16 psec/m
and the mechanical neutron monochromator): 1) gaseous
nitrogen at pressure 8.5 atm; 2) liquid, sample thickness
7.5, 10, and 29 mm. Cross section of (n, p) reaction reck-
oned on 1/v law.
We note once more that all these considerations have
a qualitative character, since we cannot in reality separate
out a part of the neutron cross section corresponding to scat-
tering without energy transfer.
The effects described above were first observed experi-
mentally in sulfur, lead, and bismuth by Chamberlain and a
little later in helium by Somers et al. [4],. Coherent processes in the scattering of neutrons in liquids without energy
exchange were first discussed in detail in [5].
Below, we present some results of measuring the total neutron cross sections of liquid oxygen and nitrogen and
gaseous nitrogen as a function of neutron wavelength in the vyR-ivi reactor of the Institute of Physics, Academy of
Sciences, UlaSSR. Nitrogen and oxygen have large coherent cross sections (11.0 and 4.2 barn respectively) and are,
therefore extremely suitable for observing coherent effects in the scattering of slow neuttons.
gNPERI MENT
The total neutron cross sections were determined from the transparencies T of the samples:
1 ,T,
t ?
nx
(7)
The liquid nitrogen was at the boiling point of 77?K. To determine its cross section, samples 7.4, 10, and 29 mm
thick were used. Measurements without sample were made by replacing the nitrogen with aluminum containers.
The liquid oxygen was at its boiling point; the sample thickness was 29 mm.
For measuring the cross section of gaseous nitrogen, two cylindrical containers with Duralumin tops were made.
The length of these was 50 cm and the diameter 60 mm. One of the containers was filled with nitrogen up to a pres-
sure of 8.5 atm, the second was pumped out to 104 mm Hg. For determining the transparency of the nitrogen, these
containers were placed in turn in the neuron beam.
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Impbarn
Nil
III
The arrangement of the experiment is indicated in Fig. 1.
For reducing the background of fast neutrons a liquid-nitrogen-
to cooled beryllium filter 9 cm in diameter and 25 cm thick was
used. The variation of the crosS section of liquid nitrogen with
wavelength was determined by the time-of-flight method by
means of a mechanical neutron interruptor and a mechanical
neutron monochromator with a longitudinal axis of rotation. The
neutrons were controlled by a battery of ten Born counters
SNMO-5 set at 3 m from the sample. The maximum height of
5
the counting space was 18.5 cm, the breadth 10 cm, and the
depth 12 cm. The flight time was measured with a multichan-
? nel analyzer with ferrite memory. In order to indicate the reso-
lution of the systems, Figs. 2 and 3 show neutron spectra after
passing through the beryllium filter. The wavelength graduation
for the mechanical monochromator was effected with respect to
0 5
gold. The cross sections of liquid nitrogen were determined for
10
various dimensions of the counting surface and with samples of
Fig. 5. Total cross section of oxygen: 1) gas [7]; various thickness (T = 0.2 to 0.75). Various methods of mono-
2) liquid; measured by means of the mechanical chromatizing the neutrons were used. The results are in mutual
monochromator. agreement.
A,A
The cross sections of liquid oxygen and gaseous nitrogen were only measured by means of the mechanical
monochromator.
RESULTS OF MEASUREMENTS AND DISCUSSION
The results of determining the cross sectionsare shown in Figs. 4 and 5. By comparing the graphs we see a
sharp difference between the behavior of the cross sections for the gases and liquids. For the liquids there is quite a
sharp fall in cross section after X P--; 5.5 A. This effect may naturally be ascribed to the existence of short-range
order in the liquids. The angular distribution of scattered neutrons with x = 1.08 A in liquid oxygen and nitrogen
was studied in [6]. In principle the value of i(s) - I (s)?I', where I(s) is the observed scattering intensity, and
I = I (s) could be obtained from these data. There are difficulties, however, in determining These were ag-
gravated by the fact that we only had graphs taken from a journal article at our disposal. The inaccuracies in deter-
mining 100 are increased on using formula [5] owing to the factor x2. We should therefore not expect any quantita-
tive agreement between the transformed results of [6] and our measured total cross sections. We made a transforma-
tion of the data for nitrogen [6] in
accordance with Eq. (5). In order to make the cross section positive, a constant
quantity differing from the constant determined from the graph
of [6] had to be added at each point. The variation of the elas-
tic-scattering cross sections so obtained with wavelength is shown
in Fig. 6. This is noticeably similar to the experimental curves.
The maxima for nitrogen in both cases lie at x ??? 5.5 A. In gen-
eral, the effect of the fall in cross section for waves corresponding
to the position of the maximum in i(s) must be expected for all
liquids in which the coherent scattering cross section is compar-
able with or much larger than the incoherent and reaction cross
sections. The fall in the neutron cross sections of D20 [7] at
0.002 eV may also be connected with the effect of coherent scat-
tering. In fact, using the angular distribution of neutrons with
X = 1.12 A scattered in heavy water measured in'[8], together
with relation (5), we can show that o c?h varies sharply for Eneut
elas
5
A,A
Fig. 6. Variation of the total elastic coherent
neutron-scattering cross section in liquid nitro-
gen with wavelength, as obtained from neutron-
diffraction data [4].
588
? 0.002 eV (X = 6.26 A).
From the energy dependence of the neutron cross sections
in the liquid phase, we can estimate the upper limit of the
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inelastic-scattering cross section. This is smaller than that predicted by the Nelkin-Krieger model. The discrepan-
cies are to a certain extent connected with the inaccuracy of this model. They may also be connected, however,
with the retardation of molecular rotation (x-ray and neutron-diffraction data indicate that oxygen and at any rate
nitrogen are molecular liquids [3, 6]). Hence, the measurement of total neutron cross sections over a wide temper-
ature and pressure range may enable us to follow the process of the retardation of molecular rotation on approaching
the condensed state.
In conclusion the authors consider it their duty to thank V. A. Gul' ko, V. F. Razbudei, V. L. Nechitailo, and
V. A Medvedev for help in the work.
LITERATURE CITED
1. L. Van Hove, Phys. Rev., 95, 249 (1954).
2. G. Vineyard, Phys. Rev., 110, 999 (1958).
3. N. Gingrich, Rev. Mod. Phys., 15, 90 (1943).
4. 0. Chamberlain, Phys. Rev., 77, 305 (1950).
5. H. Sommers et al., Phys. Rev., 97, 855 (1955); L. Goldstein, Phys. Rev., 84, 466 (1951).
6. D. Hershaw et al., Phys. Rev., 92, 1226 (1958).
7. J. Hughes, Neutron cross sections, BNL-325 (1958).
8. B. Brockhou, Nuovo cimento, 9, 45 (1958).
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TOTAL AND DIFFERENTIAL FISSION CROSS SECTIONS
OF URANIUM AND THORIUM FOR LOW ENERGY DEUTERONS
(UDC 539.172.13 + 539.17.015)
Yu. A. Nemilov, V. V. Pavlov, Yu. A. Selitskii,
S. M. Solov'ev, and V. P. gismont
Translated from Atomnaya Energiya, Vol. 18, No. 5,
pp. 456-459, May, 1965
Original article submitted June 23, 1964
A method of recording fission fragments with glasses is used for determining the total and differential
fission cross sections of Th232, U233, U235, and U238 for deuterons with energy considerably below the
Coulomb barrier-6.6 MeV and less. The results are compared with data from published papers in
which the same quantities were studied by means of semiconductor detectors in the region of high-
energy deuterons. New facts are noted, which indicate that for nuclei with different neutron fission
thresholds, as a result of irradition with sub-barrier deuterons, the reactions preceding fission may
differ considerably.
In recent years, due to the development of techniques for recording fission fragments (especially semiconduc-
tor detectors and mica [1] and glass [2] methods) the study of the fission of heavy nuclei by charged particles with
energy below the Coulomb barrier has been made possible. Thus, by means of semiconductor detectors, in [3] the
fission cross sections for Th232, U233, U235 and U238 were determined for deuterons with energies of 5.8-6.6 MeV, in
[4]?the fission cross sections of Th232 and U238 for protons and deuterons with energies 3-12 and 6-12 MeV respec-
tively, and in [5]?the cross sections and angular anisotropy were measured for the fission of Th232, U233 and U235 by
deuterons with energy 8-20 MeV.
In the present project a method of recording fission fragments with glass plates was used, which enabled the
total as well as the differential cross sections to be measured at even lower deuteron energies. Investigations with
sub-barrier deuterons are of particular interest since it was established earlie in [3] that different reactions precede
fission in different nuclei: in the even-even nuclei Th232 and U238, which have a high deuteron fission threshold,
fission is originated mainly by the complete merging of the deuterons with the target nucleus, while with nuclei
with odd number of neutrons U233, and U235, which are fissioned by thermal neutrons, fission takes place principally
as a result of capture of a neutron in a stripping reaction. In the first case, a:nucleus with high excitation but small
angular momentum is formed, since for sub-barrier deuterons central collisions are the most effective. In the sec-
ond case, on the contrary, excitation of the nucleus due to the transfer of energy to the stripping proton may turn out
to be very low, close to the fission threshold, but the moment may be relatively high [6]. The differences in parity
of the number of nucleons, the energy of excitation and the angular momentum of the fissioning nucleus should be
reflected in the probability of fission and in the angular distribution of the fisSion fragments.
PROCEDURE FOR THE INVESTIGATION
The principal complexity in investigating the fission of nuclei by charged particles consists in the fact that
such fission fragment detectors as ionization chambers, scintillation counters, :semiconductor detectors, photoemul-
sions can be overloaded by background from particles which are scattered by the target. The method of capture by
foils also has considerable disadvantages which are associated with the unavoidable reduction of the recording ef-
ficiency by the complexity and uncertainty of the chemical separation of the elements.
It was reported in [1, 2] that in the surface layer of mica or glass, fission fragments produce such significant
damage that subsequent etching by fluoric acid forms craters, easily visible under the microscope even at low mag-
nification. In the present paper, glasses were used as the most simple and most convenient method of recording
fission fragments.
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lcm
Fig. 1. Diagram of the fission cross
section measurement: 1) diaphragm;
2) foil for varying the deuteron energy;
3) target; 4) glass plate for recording
fission fragments of nuclei.
a ,cm2
108
to'
233 -
u
U235
?
232
h
,
6.0 50 60 Ed, MeV
It is desirable that there should be the smallest pos-
sible structural damage and nonuniformities on the surface
of the glass. Normal photographic plates are completely
suitable for this, protected from damage by a layer of emul-
sion, which has been carefully stripped from the functional
surface with a 3% solution of fluoric acid. The exposed
plates were etched in acid of the same concentration and
were examined under the microscope at x200 magnification.
The irradiation geometry ensured an almost perpendicular
path for the fission fragments into the glass surface (in this
case, the recording efficiency is 100%). The glasses were
put on metallic holders with openings 7 x 7 mm for fission
cross section measurements and 6 x 16 mm in the case of
measurements of the fission fragment angular distribution.
Targets of uranium and thorium, with a thickness of
? 200 pg/cm2, were prepared by evaporation in vacuo of
the fluorides [7] on thin silver backings 2 x 2 cm and with
a weight of 500 pg/cm2, stuck to the framework of a cop-
per foil. The silver films were quite stable and did not
change under the action of the beam of charged particles,
absorbing only a small fraction of the fission fragment energ
The accuracy of determining the energy of the deu-
terons, which were accelerated in a cyclotron, was 0.1 MeV.
In order not to worsen the geometrical resolving power, in
the experiments for measuring the angular distribution of
the fission fragments, the energy of the deuterons was varied
within small limits in the cyclotron itself, and in the case
of cross section measurements reduction of energy from the
primary value (6.6 MeV) was carried out by retardation in
platinum foils. The maximum half-width of the deuterons
in the energy spectrum, due to the retardation, did not ex-
ceed 2%. The deutron current passing through the target,
equal to ? 0.5 ?a, was measured with a current integrator.
y.
FISSION CROSS SECTIONS
A diagram of the experiment for measuring the fission
cross sections is shown in Fig. 1. In order to eliminate the
Fig. 2. Relationship between the fission cross section distortions in measuring the deuteron current, on the last
collimating diaphragm, located outside the influence of the
direct beam, a potential of 200 V was applied which pre-
vented the escape of secondary electrons from the surface of the filter. In the measurements undertaken in this
project, primary consideration was given to the relative nature of the dependence of the cross section on the deuteron
energy; absolute values were obtained which were normalized with respect to previous data for Ed = 6.6 MeV [3],
and which for Th232 and U238 agree with the results of [4]. As the experimental results showed, the measurement of a
fission cross section of ? 10-32 cm2 by means of glasses is not very complex, but in the case of deuteron irradiation
of nuclei the background of neutrons formed by light nuclei as a result of (d, n) reactions is a hindrance.
The effect of the neutrons from fluorine was estimated by comparing a number of targets of different weight.
The background determination in this case was based on the fact that the number of nuclear fissions by the deuterons
is proportional to the target weight, but the number of fissions by neutrons from the fluorine is proportional to the
square of the target weight. The fission cross sections of nuclei, with the background taken into account, is shown in
Fig. 2. The steeper decrease of of for U235 and U233 in the region of small deuteron energies may be connected with
a and the deuteron energy Ed.
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tk\
lcm
1.2
0.8
Fig. 3. Arrangement for
measuring the angular dis-
tribution of the fission frag-
ments: 1) diaphragm;
--a? 2) glass plate, for record-
ing fission fragments from
nuclei; 3) target; 4) Fara-
day cylinder.
I
u2"
I
.
11111aw.Millifi3
la
MI
0 30
60 .90
C. M.
degrees
S. '
120
150 180
Fig. 4.Relationship between the differential fission
cross section and the angle in the center of mass
system (c.m.s.) relative to the direction of the neu-
tron beam for Th232 and U235.
the gradual reduction of the fraction of fissions after the
(d, p) reaction because of insufficient excitation of the
nuclei for subsequent fission. It follows from the data
given below that the angular anisotropy of the fission frag-
ments is of low magnitude, and, consequently, the corre-
sponding corrections were not introduced into the cross
section.
The cross section curve for U233 (see Fig. 2) falls
somewhat more abruptly in comparison with the previous
measurements [3], which is explained, probably, by taking
a more accurate account of the background of fission neu-
trons formed in the target.
ANGULAR ANISOTROPY
In preparing the experiment, the fact was taken into
account that the anisotropy of the angular distribution of
the fission fragments may be small, and, consequently,
the geometrical resolving power must be combined with
high Statistical accuracy.
The structure and dimensions of the chamber used
for measuring the angular distribution of the fission frag-
ments are shown in Fig. 3. Thirteen glass plates allowed
the fission fragments to be recorded simultaneously at 12
angles over the range from 23? to 1700 relative to the di-
rection of motion of the deuterons with an aperture of
angle on each of the plates equal to 70. Central passage
of the beam of deuterons through the target was ensured
by collimators and was controlled by adjusting the cham-
ber and by coincidence of the number of recorded fission
fiagments on pairs of glasses located at an angle of 90? to
the direction of the beam. The number of fission frag-
ments recorded by a single plate was 2000-4000.
It was established that for all nuclei the differential fission cross sections over the range of deuteron energies
investigated vary smoothly within small limits, and, therefore, we have limited ourselves to showing the groups
corrected for the effect of motion of the center of mass,* only for Th232 and U235 for Ed = 6.6 MeV (Fig. 4), and the
remaining results are given in the table. Two facts should be noted: 1) the small anisotropy value in all cases;
2) the "anomalous" nature of the anisotropy for U235, consisting in the fact that in contrast to other nuclei, the
angular distribution of the fission fragments has a maximum not at 0?but at 90?. Both the facts noted are due to the
nature of the interactions leading to fission and in the new aspect they confirm the conclusions reached in [3]. The
low anisotropy value for Th232 and U238 is associated with the fact that fission of these nuclei by sub-barrier deuterons
takes place because of the complete coalescence of the deuteron with the target, leading to a compound nucleus with
a high energy of excitation but with a small angular momentum. Fission of U233 and U235 takes place primarily as
a result of a (d, p) stripping reaction; in this case, the anisotropy is found to be small, obviously, because the orbital
momentum of the neutron entering the nucleus (at such low deuteron energies) is also small and there is no sharp
orientation relative to the beam of deuterons. Even the excitation of the nucleus, fissioning after stripping, must be
*A correction to the impulse of fissioning nucleus was introduced in all cases on the assumption of total coalescence
of the deuteron with the target nucleus. It can be seen from Fig. 4 that for U235, the cross section at an angle of 100
to the deuteron beam was found to be somewhat larger than the cross section at an angle of 17.0?, which is possibly
associated with the momentum produced by the proton from the stripping recoiling backwards.
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Magnitude of Fission Anisotropy, More Precisely a (10?)+ a (1700)
20 (90?)
for Various Nuclei and Neutron Energies
Ed,
MeV
159U233
U23
U238
Th232
6.6
1.05?0.03
0.95?0.03
1.05?0.03
1.10?0.03
6.0
1.08?0.03
0.95?0.03
1.00?0.04
1.10?0.04
small. The anomalous anisotropy of U235 also confirms this, since the latter is possible only for a small number of
open fission channels [9-11].
On the basis of the concepts developed, it is possible to give a first opinion on the strange fact noted in [5]:
despite the normal dependence of fission anisotropy of strongly excited nucleton the parameter Z2/A, in the region
of sub-barrier deuteron energies the anisotropy for U233 was found to be not less but greater in comparison with U238
This relationship is not surprising if one takes into account that because of the difference in the reactions, excitation
of the U233 nucleus is considerably less than excitation of the U233 nucleus. Further, when fission takes place by an
identical route, for example, in the case of irradiation by sub-barrier deuterons of Th232 and U238, the anisotropy for
the nucleus with the lowest parameter Z2/A, i.e., Th232, is found to be the greater (see table), which also should be
expected.
From comparison of our data on U235 with the results of [5], it can be seen that the increase of anisotropy re-
ported for Ed = 7 MeV is associated with statistical deviations.
In conclusion, the authors express their thanks to the group of staff responsible for operation of the cyclotron.
LITERATURE CITED
1. P. Price and R. Walker, Phys. Rev. Letters, 3, 113 (1962).
2. V. P. Perelygin, S. P. Tret' yakova, and I. Z?vara, Preprint OIYaI P-1323 [in Russian], Dubna (1963).
3. G. I. Marov et al., ZhETF, 44, 1445 (1963).
4. G. Choppin, J. Meriwether, and J. Fox, Phys. Rev., 131, 2149 (1963).
5. G. Bate, R. Chaudhry, and 3. Huizenga, Phys. Rev., 131, 722 (1963).
6. H. Britt et al., Bull. Amer. Phys. Soc., S2, 8, 525 (1963).
7. A. I. Baranov et al., Pribory i tekhnika eksperimenta, No. 5, 173 (1962).
8. V. Elmor and M. Sands, Electronics in Nuclear Physics [Russian translation], Izd-vo, IL, Moscow (1953).
9. 0. Bor, in the book: "Data from the International Conference on the Peaceful Uses of Atomic Energy
(Geneva, 1955)" 2, [in Russian], Fizmatgiz, Moscow (1958), p. 175.
10. V. M. Strutinskii,?ZhETF, 30, 606 (1956).
11. L. Welets and D. Chase, Phys. Rev., 103, 1296 (1956),
All abbreviations of periodicals in the above bibliography are letter-by-letter transliter-
ations of the abbreviatious as given in the original Russian journal. Some or all of this peri-
odical literature may well be available in English translation. A complete list of the cover-to-
cover English translations appears at the back of this issue.
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AN EXACT GENERAL SOLUTION IN SPHERICAL HARMONICS
OF THE BOLTZMANN EQUATION
(UDC 621.039.51)
G. Ya. Rumyantsev
Translated from Atomnaya gnergiya, Vol. 18, No. 5,
pp. 459-463, May, 1965
Original article submitted May 18, 1964
A solution of the one-velocity kinetic Boltzmann equation is obtained as a series of spherical har-
monics. General expressions are obtained for the terms of the series, derived without any approxi-
mately valid assumptions. As particular cases of this solution, we obtain formulas for the known
pN -approximations for the spherical-harmonic method.
The exact general solution of the kinetic equation in the form of a series of spherical har-
monics contains arbitrary functions which must depend on the formulation of boundary conditions.
The general determination of the boundary conditions and the arbitrary functions is not considered.
All the results of [4] remain valid for PN -approximations.
The well-known and usually very effective spherical-harmonic method [1-6], by which the integro-differen-
tial Boltzmann equation is reduced to an infinite system of differential equations, is always applied in a PN-approx-
imation- In this approximation only the first N + 1 equations of the system are solved under the artificial condition
that the spherical harmonic of number N + 1 be identically zero. In some cases, however, this condition is too
rough, and for small values of N the spherical-harmonic method, in the form in which it is at present used, leads
to large errors.
In the present work, we give an exact solution of the infinite system of equations obtained by the spherical-
harmonic method; this solution is essentially equivalent to an exact solution of the Boltzmann equation. Our solu-
tion is general, i.e., it is not subject to any boundary conditions, and so it is determined with a precision up to cer-
tain arbitrary functions. We do not consider the formulation of boundary conditions and the solution of concrete
physical problems in the present work. Different approximate approaches to this proble.m are possible, the use of
which would obviously considerably broaden the range of applicability of the spherical-harmonic method. The or-
dinary PN -approximations together with the appropriate boundary conditions are easily obtained as special cases of
the general solution.
Harmonic Polynomials and Vector Operators
We seek the solution of the one-velocity kinetic Boltzmann equation in the form
___,_J (2n+ 1) Yn (r, 0).
n=0
(1)
All the notation is the same as in [4] except when differences are specifically mentioned. Each spherical harmonic
Yn corresponds to a homogeneous harmonic field Un:
Un (r, = RuYn (r, a). (2)
We note that Un, being a polynomial in 1.1x, py and pz, satisfies in the space p the Laplace equation
PU?O
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The coefficients of the polynomial are functions of r. We consider those polynomial satisfying the equation
v2un, (4)
in the space r, where x2 is a constant, for the moment arbitrary.
Let the operator (4V)/ denote the /-tuple application of the operator (pV). Polynomials in( irV) introduced
below will have their obvious meaning. We now demonstrate the theorem that enables us to obtain an exact ex-
pression for any term of the series (1).
Theorem. If fk is a harmonic polynomial of degree k satisfying
(VV) O; x2/h,
then the operator ti,'hP(11) ?11tV?)
[Ix
converts it into a harmonic polynomial of degree n. Here and below
(k)
Pn (t) = dk/dtkPn(t), where P(t) is a Legendre polynomial.
(5)
Proof. The operator hP(nk) ( 11V is a homogeneous polynomial of degree n?k in /ix, p ?
y, and ,, since
!Ix
it is a linear combination of expressions of the type
ILV - 2q
u 2g
)
where 0 q n?k/2. Hence ix"-q))
that this polynomial satisfies the Laplace equation:
2 ?i ( tar -NII-112q
- 7
or (Ill a-?if z
h is a homogeneous polynomial of degree n. It remains to prove
1,21171-':p(,/t) ILV
P?Y? } 1"
(6)
i.e., that it is harmonic.
This is proved by induction. We first show that Eq. (6), for a fixed k, is satisfied for any value of n if it holds
for n-1 and n-2.
The operational relations needed for the proof are the following:
(PS7)=--(11S7)24- 2 (0);
t,t2V2_,1_, 4 6:
'(V i7) 1-1,2h (11V )1= 2k (I VY 07
(tiv) a [It2h (11v)`1v2 a EFL21' (a)11
(7)
(8)
(9)
OIL O(iv)
We also need the following recurrence relations for derivatives of the Legendre polynomials*:
(n? k) 13) (t) = (2n ?1) tPri (t)? (n k 1.).1)(P 2 (t);
(10)
(n ? k ? 1) tP(nki (t) (1 ? t2) Pnk_til) (t) (n + k? 1) 13(7,ki 2
(t).
(11)
For brevity we use the notation
p(Th.) ?IV = p(k) il(h)
(12)
(Sr
[ix n ) ? n ?
Using (10), we write
(nh )n 1 1) I f)( -- (n+ k ?1) 2
- 2
?
(13)
nk-)
k (2n p,2to
X
*It is possible that (11) is not in the literature. It can be derived from other known recurrence relations.
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Now using (7) and (8) and assuming that (6) holds for n-1 and n-2, we obtain
?
V210-hiVniofh, 1 f(2n ?1) 2 (v) tin
n?k 1
To an operator of the form (VV) tAn-11? I K) I we apply the formula (9) and, using (5), we have
(Vi7)p.V y pcnk-i-in} fh
n n 1 Vnk?' f
1 h = 2 X f(n k 1) (11?) Kil) 4-- [1 ?
--1 -
x 1-tx
or, from (11),
(VG)
n? k? p(h) li4
n?k =(fl-I-k?I) fin- h? 2 Pnk) 24.
X
. (14)
(15)
(16)
We substitute the last expression in the right-hand side of (14) and eliminate the operator (liP) by using the Euler
identity:
(iv)Rn?h-2Kk?
21k = (n-2) lin-h-43(:)21k ? (17)
(17)
The relation (17) is a consequence of the fact that the expression 1.0?h? 2P(rth) 2ik is a homogeneous polynomial of
degree n-2.
Using (16) and (17),we find that
.V21.12/?ilii(nk)fh _= 0.
Hence, for the above assumptions a homogeneous polynomial of degree n of the form jun?hP(?k)fh is harmonic.
We have n k in all expressions, and so for the completion of the proof, it is sufficient to prove that (18) is
satisfied for n = k and n = k + 1. This is done directly:
for n = k
(18)
for n = k + 1
?- (to
V2Ph fk= const 2fi,=-
?21-11V,-ith=const V2 (jAV) fa = const [(.117) 2 (VV)1 fa =0.
Consequences of the theorem. Every harmonic polynomial Un, satisfying the equation
v2un x2un,
Can be written as a sum:
Un = 1,1,n-htV
h=0
It is easily shown that the terms of this sum cannot be linearly dependent, and that the sum is a harmonic
polynomial of general type.
Setting bt = 1 in (20), we obtain an expression for the spherical harmonic Yn:
n
where yk = 01.,:=4 is a spherical harmonic of order k.
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(20)
(21)
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The Solution of the Kinetic Equation
The solution of the homogeneous equation is of special interest and so, because of the brevity of this article,
we will consider the one-velocity kinetic Boltzmann equation only for a source-free medium:
(S2V)F (r, + EF (r, 9) ? F (r, 52') W (S2' 51)6/52' _?: 0.
Substituting the function F (r, 52) in the form (1) into Eq. (22), we obtain
(22)
E (2 1) f(S/V) Y0 H- =0, (23)
n
where En = E? Escn and the cn are the coefficients in the expansion of the scattering indicatrix (for more details
see [4D*
We now express Yn in the form (21) and use (10). In order not to mix functions yk belonging to different har-
monics Yn, we use for them the notation ynk, and remember in the following that the order of the spherical harm-
onics ynk(r, 62) is determined only by the second subscript. Then (23) becomes
CO n
k H- I) 1Y?h2_1 - (n k) P 1] Ya -1,-? (2n -.1r 1) 1-:?][.).Y,:h} =0.
nr=0 h=.0
If the linearly independent terms of the expression?in (24) are equated to zero, we obtain
x Rn-- k) y, h --H(n k hi (2n + 1) Iny?k = 0.
This relation enables us to express all the functions ynk in terms of ykk. We assume that
!lot= (-1)'' I'MnitYkk,
(24)
(25)
(26)
where we obviously have Mkk = 1.
We now introduce the notation v = E/x and en = En/E. The relation (25) now yields the recurrence relation
(n + I) M ?+1,a = (2n + 1) venMi,/,? (n ? (27)
for the coefficients Mnk, and so these coefficients are polynomials of degree n?k in v, where each of these poly-
nomials contain either purely even or purely odd powers of u.
In (25) and (27) there are quantities with the same subscripts k, and so for each k there is an independent in-
finite sequence of functions ynk beginning with the function ykk. In the general solution the functions ykk are ar-
bitrary except for the fact that, from the assumption, they must satisfy the equations
(0) fil'yhh= 0;
( E
V Yah =
Yhh?
(28)
(29)
In concrete problems, the form of the functions ykk is determined by the geometry of the medium and the boundary
conditions. Substituting (26) in (21), we obtain
Yn (r, Q, v)= E ( - (v) iiNkh(r, 0, V).
h=-0
*In [4] En = (2n + 1) (E? Escn).
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All real angular distributions are such that the functions Yn in the expansion (1) cannot increase indefinitely
.with increasing n. We require that the coefficients Mnk(v) also satisfy this condition. An investigation of the rela-
tion (27) shows that the functions Mnk(v) are bounded for all real v in the region 11 I < 1 and for certain values
v = ? vk in I v I > 1. Hence, the general expression for the spherical harmonic of order n must be
Yn(r, 0) = (?i)fl_k fm,?3,(voi'Vykk(r, 0, vh)+ M nh (V) iliik)Y (r, v) dv ;
h=0 0
(31)
The integration here is only for positive values of v, since the general form of the functions ykk for the complete so-
lution in the large is independent of the sign of v. The numbers ilk can always be given a plus sign. The values of
the ilk depend only on the physical properties of the medium. In each concrete case, the vk can in principle be cal-
culated with any desired degree of accuracy.
PN-Approximations of the Spherical-Harmonic Method
As a special case of (31), we can obtain formulas for the PN-approximations of the spherical-harmonic method,
if, without worrying about the convergence of the Mnk(v) when n co, we specify the conditions
MN+1. k (V) = 0 0 < k (32)
and yN +17 0, which clearly agree with the assumption YN+1 0. Then the set of permissible values of v reduces
to the set of roots vki of Eq. (32).
The roots voi correspond to those numbers -rtoi which in [4] were called fundamental numbers, and the roots
vki for k> 0 correspond to the supplementary numbers Rki. In one-dimensional, plane, or spherical geometry, the
functions ykk for k > 0 must be identically zero, since otherwise it is impossible to satisfy the condition (28), i.e.,
the supplementary roots take no part in the solution of such problems.
In the general case, among the roots vki there will also be zero roots of the odd polynomials MN +1, k. The
corresponding functions ykk(r, 5), 0), if they are defined to be
Ykk (r, 9, 0) =urnYak (r, v),
v-o
have an infinitely small relaxation distance, and their contribution to the general solution is in the form of finite
discontinuities on the boundaries. It can be seen that for even N the discontinuities (or jumps) will occur in even
harmonics, and for odd N in odd harmonics; the component of the first harmonic which can be interpreted as the
normal component of the diffusion current, however, is always continuous.
The existence of zero roots was also noted in [3, 6], but these roots and the corresponding special solution
were discarded. In [4], where the PN-approximations of the spherical-harmonic method are described in their most
general form, the zero roots are also not taken into account, but in this work the discontinuity of certain spherical
harmonics follows from the boundary conditions. When zero roots are taken into account, the requirement that all
the spherical harmonics be separately coupled becomes formally realizable. It can be shown that in the PN -approx-
imation, because of the above-mentioned properties of the functions ykk (r, l, 0), the formal coupling of all harm-
onics is actually equivalent to the satisfaction of the boundary conditions given in [4].
CONCLUSIONS
The exact general expression for harmonics of any number on the one hand simplifies the analytical transition
to PN -approximations of the spherical-harmonic method. This simplification is particularly important in multidi-
mensional problems. As is clear from the preceding, PN-approximations are obtained by approximating the infinite
set of .characteristic numbers v and functions ykk(r, v) by a finite set of numbers vki and the corresponding func-
tions ykk(r, l, i). The boundary conditions derived in [4] remain in force, but they have a new mathematical
interpretation.
On the other hand the assumption YN+1 = 0, which lies at the basis of the usually employed PN -approxima-
tions, is not obligatory here. It is thus possible that new approximate forms of the spherical-harmonic method may
598
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CALCULATION OF WEAK SELF-OSCILLATORY
CONDITIONS IN NUCLEAR REACTORS
(UDC 621.039.51)
B. Z. Torlin
Translated from Atomnaya Energiya, Vol. 18, No. 5,
pp. 463-468, May, 1965
Original article submitted February 24, 1964; final revision submitted July 17, 1964
A method for the approximate calculation of the parameters of self-oscillatory conditions in nuclear
reactors with small power oscillation amplitudes is presented. An equation is derived, which, within
the framework of the assumptions made, can be used for determining the frequencies of the reactor's
possible self-oscillatory states as well as the amplitudes of reactivity and power fluctuations and the
phase difference between these fluctuations.
The first problem that is ordinarily encountered in investigating the dynamics of reactors with feedback is the
problem of the system's stability. If sufficiently small deviations from the equilibrium state are considered, the so-
lution can be obtained in a relatively simple manner by using the known methods of linear analysis. If the system
is stable, its deviation or the amplitude of its deviation from the equilibrium state will asymptotically tend to zero
after any sufficiently small disturbance. If the system is unstable, linear analysis cannot yield any information on
its asymptotic behavior. At the same time, beyond the limits of the linear stability region, a reactor can actually
have very interesting stable self-oscillatory states with very small amplitudes. Actually, if the oscillation ampli-
tude does not exceed, for instance, the noise level in the system, this implies a broader stability region, and, con-
sequently, a wider range of the reactor's operating conditions.
The above apparently does not constitute all the useful information that can be obtained as a result of solving
the problem of stable self-oscillatory conditions with a small oscillation amplitude in a reactor.
We shall use equations of the following types for considering the mean neutron density N throughout the reac-
tor's volume for the mean concentrations ci of sources of all the m groups of delayed neutrons.
dci
cit
171
dN R Q ?1 N kici;
dt =r T
i=1
1=1, 2, . . . ,
T '
(1)
where pis the system's reactivity in dollars. T is the mean lifetime of neutrons, ai and Xj are the yield and the de-
cay constant of the j-th delayed neutron group, respectively, and 13 =.E 13.,. For the investigation of small devia-
tions of the neutron density from a certain theoretical value No, we shall introduce the notation
N=N(1-1-n) (2)
(1\1. is the mean value of the neutron density, which is generally different from No), and we shall reduce system (1) to
the following form
600
w1(P)n?Q(1-r n), (3)
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T vim co
where p = d/dt is the differentiating operator, while w1(p) = p(T + ) is the linear operator, (4)
where Pi
a.; .
For the sake of simplicity, we shall subsequently investigate a system with linear feedback* :
w2 (P) n No = Q'
(5)
where w2(p) is the linear operator; the N/No coefficient appears as a result of the fact that w2(p) is deterMined for
the theoretical neutron density No, while n pertains to the mean neutron density N.
Since we are interested only in he periodic solutions of the system, the problem can be divided into two parts:
1. Finding of the periodic solutions n(cot) of the nonlinear Eq. (3) for the assigned periodic action Q (wt). It
is found in this case that the Q (wt) function must satisfy certain requirements if Eq. (3) is to have periodic solutions
n (wt).
2. Determination of such q (wt) values for which the corresponding functions n(u)t) would yield the w2(p)
function after the Q(wt) operator is applied to them.
The solution of the first problem for I n I < 1 can readily be obtained by using the method of successive ap-
proximations [1]. If the Q (wt) function is given by
CO
Q (0) = ao E aheikot; k 0
?co
and n(cot) is sought in the following form:
the expression
is given by
,W I
to (cot) -=
Aleihot; 1 0,
for every subsequent + 1) approximation in terms of the preceding (ii-) approximation
a/ E akm,v),
A(i?)+1)_-'
Lei (h)l)
In order to secure a periodic solution of Eq. (3), it is necessary to satisfy the condition
00
ao ? 2Re a124_1
1=1
or, when using the method of successive approximations
a2+= -
( 0 2Re
1=1
*The same method can also be used in the case where the feedback equations contain nonlinear terms.
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(6)
(7)
(8)
(9)
601
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The linear approximation is used as the zero approximation.
The second part of the problem can also be solved by using the method of successive approximations. In this
case, it is natural to use expression (6) for the first approximation, retaining in it only the first harmonic. The sub-
sequent approximations for Q(wt) can be obtained by substituting the corresponding solutions n(wt) of Eq. (3) in ex-
pression (5).
It should be remembered that, in order to secure a negative reactivity shift ao, the mean neutron density 171 in
the reactor must deviate from the calculated steady-state value No. The relationship between these values is
given by
N = No (1 ? x0a0),
where xo is the negative feedback gain at the "zero frequency."
By using (8) and (10), we readily obtain the following expressions in the second approximation:
A i( 2 (ico) fl -1- ai 12Wil (i(o) [wi-1(2i(0)? 2 Re w;' (io.))1 -1- ai 1401 (ico));
A 2( 2 = ai2wi-1(i(0) wil (2i(0) [1 ? at 1202 (1(0)].
(13)
Expression (13) indicates that oscillations with a frequency twice as high as the frequency of the basic disturb-
ance arise in the system. Through feedback (5), the oscillations with this frequency will begin to act on n in Eq. (3).
The contribution of this effect to the expression A1(2) is
wi-I(Uo)w,-2 (24.o) w2 (21(o) a, 1 a1 12. (14)
If the value of this term is of the same order as the rejected terms, it can be neglected. In practice, this can be
done if I w2(2ico) I x2.
Useful information on the behavior of the amplitude of the periodic solution near the boundary of the linear
stability region can be obtained by means of the expression
daj2 2K 1
de K +1 xi? x2 (30)
for 0= K + 1/K, whence, it follows that, if xj. > x2, i.e., if K < K1 is determined from the relationship
*X ==3K1 ,
ift
(31)
the amplitude of the self-oscillatory conditions must gradually increase beyond the boundaries of the linear stability
region at increasingly greater distances from the boundary (the so-called soft operating conditions). In the opposite
case, the so-called hard operating conditions prevail. In this, within the boundaries of the linear stability region,
the probability of self-oscillatory conditions with an already considerable amplitude increases as the distance to
the boundary diminishes.
The other characteristic curve in Fig. 1 is the curve a, which corresponds to 09. In this case, I ail
also holds. The equation of this curve is given by
0 4K-2 3
b4=4K-3; w (32)
Region II (see Fig. 3), which is enclosed between the curves a and b, is characterized by the fact that only a
single oscillatory state with a small amplitude is possible at every point within this region. To the left of this region
I (see Figs. land 2), in which self-oscillatory conditions with small amplitudes do not exist at all.
604
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Declassified and Approved For Release 2013/09/24: CIA-RDP10-02196R000700010005-2
a
.4
3
2
MELIIIMME
52
27
LIKINFAIN
=Mil1
0
,
WM111
ri---="magam",
5:s
0 2 3 4 5 6 7 'r
Fig. 2. Linear stability region (shaded area) and the character-
istic curves in a K?T plot. a) Boundary of the region of self-
oscillatory states with small amplitudes; b) boundary of the lin-
ear stability region; c) straight line passing through the mini-
mum T values of the curves of equal roots (similar to the dashed
curves in Fig. 1).
N,
No
Ncri
r
.(