SOVIET ATOMIC ENERGY VOL. 48, NO. 6

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I JJI\ VVJY'JJ I *
Russian Original Vol. 48, No. 6, June, 1980
December, 1980
SATEAZ 48(6) 353-422 (1980)
SOVIET
ATOMIC
ENERGY
ATOMHAH 3HEPf NA
(ATOMNAYA ENERGIYA)
n
TRANSLATED FROM RUSSIAN
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Soviet Atomic Energy is 'a translation o~ Atomnaya Energiya, a
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SOVIET- ATOMIC ENERGY
A translation ofAtomnaya Energiya
December, 1980
Volume 48, Number 6 June, 1980
CONTENTS
Engl./Russ.
Comparison of Calculations of a Two-Dimensional Model of a Fast Reactor
- A. I. Voropaev, O. P. Chukhlova, A. A. Van'kov, L. N. Kudryashov,
and R. V. Nikol'skii ......... .............................:.. 353. 355
Procedure for Calculation of Fuel Depletion to Determine the Physical
Characteristics of a Fast Power Reactor in the Steady-State Mode
- G. B. Usynin and V. A. Chirkov .................................. 356 357
Power Distribution Monitoring and Control for a RBMK Reactor.
- I. Ya. Emel'yanov, V. V. Postnikov, and Yu. I. Volod'ko . ..... ........ 360 360
Analysis of Thermohydraulic Stability in Channels of a Boiling Reactor
- V. N. Smolin, S. V. Shpanskii, V. I. Esikov, T. K. Sedova,
and V. P. Shishov ............................................ 366 366
Approximate Methods of Calculating Neutron Distributions - S. S. Gorodkov ......... 372 370
Numerical Study of the Distribution from a Pulsed Source of 14-MeV
Neutrons in an Infinite Two-Layer Cylindrical Medium
-A. A. Morozov, R. A. Rezvanov, and A. I. Khisamutdinov ................ 377 374
Determination of the Details of the Resonance Structure of Both the Total Cross
Section and the Fission Cross Section of 235U and 239Pu for 2-eV-20-keV
Neutrons - A. A. Van'kov, Yu. V. Grigor' ev, V. F. Ukraintsev,
T. Bakalov, G. Ilchev, S. Toshkov, Chan-Khan'-Mai, and N. Yaneva ........... 381 377
Waveguide Probing Used for the Dosimetry of Bremsstrahlung in High-Current
Accelerators - Yu. P. Bakulin, A. P. Korotov Kikh, and N. N. Morozov ........ 386 381
A Method for Determining Isotope Composition by Coulomb Excitation of Nuclei
- V. N. Bugrov, V. V. Kamanin, and S. A. Karamyan ..................... 389 383
.LETTERS TO THE EDITOR
Models for E1 Center Accumulation in Quartz in Uranium Ore
-B. M. Moiseev and M. V. Petropavlov .............................
Three-Dimensional Calculations of a Heterogeneous Reactor - B. P. Kochurov
and V. M. Malofeev . .................................. > ... , . .
Use of a Low-Energy Accelerator for Element Analysis on the Basis of Proton-
Excited X Rays -B. A. D'yachkov, G. V. Kazantsev, and V. Ya. Pavlov........ .
Investigation of the Phonon Current in Neutron Ionization Chambers
- V. P. Ivanov, E. K. Malyshev, R. A. Milovanova, A. A. Sysoev, and
392 386----
395 387
398 389
P. N. Chistyakov ............................................. 401 391
Physicochemical Properties of Irradiated Inorganic Compounds of Lithium:
Oxide, Aluminate, and Silicates - V. G. Vasil'ev, S. R. Borisov,
N. N. Ryazantseva, and A. A. Vashman .............................. 404 392
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CONTENTS
(continued)
Engl./Russ.
Theoretical Studies on the Accumulation of 232U, 236Pu, and 238Pu in the Breeding
Zones of Hybrid and Fast Reactors - Yu. G. Bobkov, V. G. Ilyunin,
V. M. Murogov, M. F. Troyanov, L. N. Usachev, A. G. Tsikunov,
I. Kh. Ganev, A. D. Zhirnov, L. V. Tochenyi, and A. N. Shmelev ........... . . 407 395
Mossbauer Spectroscopy in the Phase-Composition Analysis of Corrosion
Deposits in a Nuclear Power Station - V. M. Sedov, P. G. Krutikov,
E. A. Konstantinov, V. A. Shishkunov, and A. A. Afanas'ev................. 409 396
Thermoelectric Properties of ZrC, UC-ZrC, and UC-UN at 285-450?K
- L. I. Gomozov and I. Sh. Akhmedzyanov ........................... 413 399
Nuclear-Physics Determination of the Steam Content in a Reactor at Kursk
Nuclear Power Station - V. I. Kulikov, S. S. Lomakin, Yu. N. Filimontsev,
V. V. Karnaukhov, Yu. M. Krutogin, A. M. Gryaznov, and V. V. Volkov ........ 416 400
Measurement of 236U Fission Product Yields in a Fast Reactor
-A. N. Gudkov, V. M. Zhivun, A. V. Zvonarev, V. V. Kovalenko,
A. B. Koldobskii, Yu. F. Koleganov, V. M. Kolobashkin, V. G. Liforov,
N. S. Piven', V. A. Tolstikov, and A. 0. Tipunkov ...................... 418 401
Yields of 165Tu, 166Tu, 167Tu, 168Tu, and 170Tu in Reactions with Protons,
Deuterons, and a Particles - P. P. Dmitriev, G. A. Molin, and M. V. Panarin ... 419 402
The Russian press date (podpisano k pechati) of this issue was 5/22/1980.
Publication therefore did not occur prior to this date, but must be assumed
to have taken place reasonably soon thereafter.
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ARTICLES
A. I. Voropaev, 0. P. Chukhlova,
A. A. Van'kov, L. N. Kudryashov,
and R. V. Nikol'skii
The specification of a two-dimensional (R-Z) test model of a fast reactor with a power of 1200 MW (el.)
was developed in 1976 by the European -American Committee of Reactor Physics (NEACRP) for comparison
of the basic physical characteristics obtained in the different countries which determine the economics and
safety of planned commercial fast reactors with sodium coolant [1].
The active zone with 10-m3 volume contains two equally large subzones of different enrichment. The
height of the active zone is 102 cm. The volume fractions of fuel, sodium, and steel are 41, 38, and 21%. The
fuel is a homogeneous U02 - Pu02 mixture. The isotopic composition of the plutonium is as follows: 67.8%
239Pu, 19.4% 240pu, 10.3% 241Pu, and 2.5% 242Pu. The thickness of the end shield is 33 cm and of the lateral
shield - 47 cm. The moist material is the oxide of dumped uranium. The computational model corresponds
to the start of reactor operation at power, i.e., no fission products and plutonium in the shields.
A systematic discussion of the calculations which have come in from the different countries has been
carried out at the English National Laboratory [2]. The results presented by the USSR of calculations of this
test model using the BNAB-70 system of constants are reviewed in [3]. A more detailed description of the
model and computational details is also given there.
At the present time the new BNAB-78 system of constants [4], which is based on recent estimates of
microscopic nuclear data and which are in agreement with the results of fundamental experiments on neutron
balance in uranium and plutonium critical assemblies, is starting to be widely used in the computational
practice of fast reactors in the USSR. Therefore, test calculations using this system of constants and a com-
parison of them with analogous calculations using other systems are of interest. Such a comparison was made
[5, 6] for a spherical test model of a reactor. Results are presented below for a two-dimensional test model
of a large breeder reactor (the NEACRP model).
The results given of calculations using BNAB-70 and BNAB-78 are obtained with the help of an NF-6
computer system [7] in which are implemented specific algorithms for preparation of the group macroscopic
constants. Taking the effect of resonant self-screening into account and the calculation of the moderation cross
section are done in specified approximations. The error due to these approximations is estimated (e.g., see
[8]) to have these values: ?0.5%o for keff, ?0.03% for the physical breeding ratio of the reactor (BR) and the
breeding ratio of the active zone (BRA), and ?10% for the reactivity coefficients.
The results of the calculations of the test model carried out in the USSR using the BNAB-70 and BNAB-78
systems of constants are compared with the calculations of other countries in Table 1 and Fig. 1. The param-
eters BR and BRA are found from the neutron balance corresponding to a conditionally critical calculation,
since the composition and dimensions of the reactor are fixed by the conditions of the specification. The fol-
lowing definitions are adopted:
CT+C11
(C+F)r9 -F-(C+F)r41 HI-(C+F) .
C
+C
AZ
AZ (2)
BRA =
(C+F)i+(C+F)r1+(C+P)I '
where C and F are integrals over the volume of the capture and fission rates in the active zone (AZ) or in the
entire reactor (r). The superscripts "5", "8", "9", "40", and "41" refer to the isotopes 235U, 238U, 239Pu, 24opu,
and 241Pu. The numbers in parentheses indicated in Table 1 are the results of recalculations of BR and BRA
Translated from Atomnaya Energiya, Vol. 48, No. 6, pp. 355-357, June, 1980. Original article submitted
October 15, 1979.
0038-531X/80/4806-0353 $07.50 ?1980 Plenum Publishing Corporation 353
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I
11
19
F
1. o
ol
l
20 40 60 80 100 120 140 160 180 200 220 240
Radius, cm
Fig. 1. Distribution of the heat-generation density
in the central plane of the reactor:, I) zone of small
enrichment; II) zone of large enrichment; III) lateral
shield; and IV) reflector made out of steel and sodium;
-) USSR (BNAB-78), ---) USA, and 0) France.
TABLE 1. Comparison of Calculations of the Basic Physical Characteristics of the Test Model
of a Fast Reactor
Criticality and breeding
parameters
Eff. of central
boron rod, ?1?
keff
with
respect
to
fuel
ith
respect
to
sodium
Effect of sodium removal,
7o keff
from
zone
small
enrich
ment
from entire
active zone
and end
shield
from entire
active zone
and end
shield of a
"hot" reactorl
Variation of keff upon heating
of the fuel from 1100?K to
2200?K, 01o
orig. com-
position
upon removal
of sodium from
active zone
and end shield
USSR
BNAB-70
1,009
1,48 (1,50)
1,06 (1,07)
-0,47
-0,39
1,9
1,7
2,0
-0,65
-0,34
BNAB-78
1,012
1,39 (1,42)
0,98 (0,99)
-0,37
-0,31
2,1
2,0
2,2
-0,58
-0,36
England
1,022
1,36 (1,42)
0,95 (0,97)
-0,40
-0,33
2,1
2,1
2,4
-0,74
-0,45
S.A.
0,993
1,39 (1,37)
0,99 (0,98)
-0,33
-0 , 28
2,4
2,4
2,7
-0,71
-0,46
rance
1,013
1,39 (1,42)
0,98 (0,99)
-0,38
-0,36
2,2
2,2
2,5
-0,71
-0,48
West Germany
1,024
1,36 (1,42)
0,96 (0,98)
-0,40
-0,34
2,0
1,9
2,1
-0,65
-0,42
Japan
1,014
1,36 (1,39)
0,96 (0,97)
-0,32
-0,26
2,4
2,5
2,7
-0,78
-0, 53
Av. value for
1,013
1,37 (1,40)
0,97(0,98)
-0,37
-0,31
2,3
2,2
2,5
-0,72
-0,47
foreign labs.
Remark: The numbers in parentheses give the result for the conditions of a critical reactor
(see text).
for the conditions of a critical reactor with a change in the enrichment. Sensitivity coefficients calculated from
generalized perturbation theory were used for the recalculation. The sensitivity coefficients are obtained
under the condition of preserving the isotopic composition of the plutonium and the enrichment ratios in both
subzones of the active zone [9].
In connection with the comparison of the efficiency of the central boron rod a variation of keff is indicated
for the two cases in which the fuel in the central region of the reactor (R = 7.2 cm, H = HAZ) is replaced by a
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TABLE 2. Comparison of Estimates of the
Error in Calculating the Physical Param-
eters of a Large Fast Reactor
Parameter
Based on errors
in nuclear data
Max. scatter
of test calc.
results
keff
+2 %
2,7 %
BR
?0,06
0,05
Eff. of boron rod
+10 %
20 %
Removal of sodium
?30 %
30 %
Heating of fuel
+30 %
30 %
Heat generation per
5 %
10
%
unit volume of active
?
zone
boron rod and in which the boron rod is placed in the central region filled with sodium. The computational
results of the variation of keff upon removal of sodium from different parts of the reactor are also given here.
The cross sections of uranium and plutonium isotopes at 2200?K (T = 1100?K for all the preceding alternatives)
correspond to a "hot" reactor.
Furthermore, the calculations of the variation of keff upon heating of the active zone from 1100 to 2200?K
are compared in Table 1. The numbers in the next-to-last column correspond to the original composition, and
those in the last column correspond to a reactor from whose active zone and end shield the sodium has been
removed.
The distribution of the heat-generation density (without y radiation taken into account) in the central
plane of the reactor is shown in Fig. 1. The calculation using BNAB-70 practically coincides with that using
BNAB-78 (maximum difference does not exceed 3%). The results of the American and French calculations are
the bounding ones between which the results of the other countries are located.
On the basis of the above one can draw the following conclusions:
1. The transition from the BNAB-70 to the BNAB-78 system of constants has resulted in a decrease of
BR by 0.08, which is produced entirely by a decrease in breeding in the active zone. The effective multiplica-
tion factor underwent an insignificant change (+0.3% keff). The efficiency of the central boron rod decreased
by 25%. The sodium void effect became more positive. The changes in BR, BRA, and keff are in good agree-
ment with the conclusions which follow from an analysis of experiments on large plutonium assemblies.[10, 11]
and with the results of international test calculations of a one-dimensional model of a fast reactor with an
active-zone volume of 2.5 m3 [5, 6]. We note that the good agreement of the criticality of the two-dimensional
model in the calculations based on the BNAB-70 and BNAB-78 systems is a consequence of compensation of an
overestimated contribution of 241 pu (-1.5%o keff) and an underestimated contribution of 238U and 239pu to the
reactivity in the BNAB-70 system of constants.
.2. Calculations of the values of keff, BR, BRA, the efficiency of the boron rod, and the sodium void effect
using the BNAB-78 system agree better with the results of foreign laboratories than the calcuations using the
BNAB-70 system. However, the change in the reactivity upon heating of the fuel is less by 25% than all the
foreign data in the calculations using the BNAB-78 system.
3. The estimates of the error in calculating the basic physical parameters of a large fast reactor ob-
tained on the basis of an analysis of the errors existing today in the microscopic and integrated nuclear data
[8] are compared in Table 2 with the scatter of the computational results in the test model under discussion.
The closeness of these and the other results indicates that the error estimates of basic reactor parameters
given in [8) are realistic. One can use them in design calculations of industrial fast reactors with oxide fuel
and sodium coolant. However, one should anticipate increases by a factor of two to three in the computational
errors of these parameters for reactors with a different coolant, a different kind of fuel, or with a design of
the placement scheme of the breeding zones which is different from the traditional one (9]. The cause for this
lies in the fact that the systems of constants used in design reactor calculations are matched for the best des-
cription of experiments on critical assemblies and operating breeder reactors with the traditional. composition.
The authors are grateful to M. F. Troyanov for his support and discussion of the research at every stage.
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LITERATURE CITED
1. C. Till, L. LeSage, and D. Wade, "Specifications for an international comparison calculation of a large
sodium-cooled fast breeder reactor," Tech. Note, ANL (Aug., 1976).
2. L. LeSage et al., in: Proc. of ANS Topical Conf. on Advances in Reactor Physics, Gatlinburg, 10-12
Apr. 1978.
3. O. P. Chukhlova et al., Preprint FEI-802, Obninsk (1977).
4. L. P. Abagyan et al., At. Energ., 48, No. 2, 117 (1980).
5. A. I. Voropaev, A. A. Van'kov, and A. M. Tsibulya, op. cit., 45, No. 6, 119 (1978).
6. A. I. Voropaev, A. A. Van'kov, and A. M. Tsibulya, op. cit., 47, No. 4, 274 (1979).
7. M. N. Zizin, O. A. Savochkina, and O. P. Chukhlova, Preprint P-40 (334), NIIAR, Dmitrovgrad (1977).
8. A. A. Van'kov, A. I. Voropaev, and L. N. Yurova, Analysis of a Reactor Physics Experiment [in Russian],
Atomizdat, Moscow (1977).
9. R. V. Nikol'skii et al., "Prediction of the physical characteristics of prospective active zones of fast
reactors on the basis of an analysis of critical assemblies and standard computational models," Lecture
at the International Symposium of MAGATE on the Physics of Fast Reactors, Ekstan-Provence, 24-28
September 1979.
10. A. I. Voropaev et al., in: Problems of Nuclear Science and Technology [in Russian], Physical Constants
Series, No. 20, Ch. 2, Atomizdat, Moscow (1975), p. 112.
11. A. I. Voropaev et al., op. cit., No. 25, 69 (1977).
PROCEDURE FOR CALCULATION OF FUEL DEPLETION
TO DETERMINE THE PHYSICAL CHARACTERISTICS OF
A FAST POWER REACTOR IN THE STEADY-STATE MODE
The breeder properties of a fast reactor determine its physical characteristics: the breeding ratio BR,
the doubling time T2, the growth rate w, the specific load of plutonium into the fuel cycle G, and the specific
amount of excess plutonium r [1]. Since these characteristics depend on the instantaneous composition of the
fuel, a unified approach to its computation is advisable. One can determine the listed characteristics for a
reactor in a constant state (continuous overload conditions; the operational reserve of reactivity is equal to
zero). We select for the calculation two different fuel conditions of the operation of a fast reactor: the steady-
state fuel mode, in which the composition of the loaded and discharged fuel is constant, and the natural fuel
mode, which is a particular case of the steady-state mode in which the composition of the loaded and discharged
fuel is identical.
Natural Mode. Let us assume that the reactor is divided into J zones. We write the equations of material
balance between the loaded and discharged fuel for the four plutonium isotopes (239-242Pu):
o
Yi Ti pl.i(Ti)(1-si)=Bz> Ti P9 .. (1)
~=1 7=1
Here ?j and Tj are the mass of heavy atoms and the fuel delay time (operating period) in the j-th zone, p?
and p i , j are the relative concentrations of the i-th isotope in the loaded and discharged fuel, and Bi is a 113
coefficient which is equal to the ratio of the discharge rate of the i-th isotope to its.loading rate. In the natural
mode Bi for all four isotopes are equal to one another (B = Bi); ej are the fuel losses from the j-th zone in
chemical reprocessing.
We obtain from the solution of the depletion equations [2]
242
Pi.i(t)_ hh,i(t)P'
h=238
Translated from Atomnaya Energiya, Vol. 48, No. 6, pp. 357-360, June, 1980. Original article submitted
August 20, 1979.
356 0038-531X/80/4806-0356 $07.50 ?1980 Plenum Publishing Corporation
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TABLE 1. Relative Concentrationof Isotopes of the Reference Reactor *
2382)
239Pu
210Pu
241Pu
242Pu
Fission products
0
i
k{
39
P
8236
0
7768
0
0,7317
0
0,9885
0,9772
,
1394
0
,
1297
0
0,1207
0,0103
0,02
,
0305
0
,
0353
0
0,0391
0,1014
0,2186
0,03147
0,0004
,
0050
0
,
0,0058
0,0067
1,7784
0,03612
0,053
0,0411
,
0014
0
0016
0
0,0019
0,1024
0,0100
0
0
,
0
,
0,0509
0,10000
-0,0513
0,02101
0,02238
*A spherical reactor with a uniform reactive zone surround by a reflector. The radius of the
active zone Ra.z.= 84.196 cm; the thickness of the shield ARs = 4.5.72 cm; the mass of heavy
atoms in the active zone ?a.z?= 7113 kg and in the shield ps = 32015 kg; the operating period
of the active zone (at an average specific power of 400 kW/liter) is Ta,z,= 727.6 days and of
the shield Ts = 1.38Ta.z, days.
0,075. 0,10 41Z5
Png, a
04 0,6
p240 0 0 /p239
Fig. 2
Fig. 1. Dependence of (EBR1 + 1); B, r, G, T2 on the depth of fuel depletion in the active zone
of a reactor in the natural mode (:;j = 0, P239,s = 0.02).
Fig. 2. Dependence of the excess breeding ratio EBR1 on the composition of the plutonium
consumed (cj = 0, P239,s = 0.02, and EBR1 = const).
where fik,j(t) are known functions of the cross sections and the time. The time t = Tj for pq,j(Tj) and pi,j(Tj) _
T
(1/Ti) J Pi,i(t) dt is selected on the basis of the established fuel depletion (active zone) or plutonium accumula-
0
tion (shield).
The criticality equation for the reactor and the conditions of constancy of the mass of heavy atoms in
each zone are written in the form
J 242 r l 1
[ fAjPi, i \ sk ?jPng, i ( kk )n , i 1 SCr
i i=238
242
P9 1;
{=238
242
P-j+Png,;=1;
i=238
242 _ _
Pi,i + Png,i = 1,
i=238
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where (6k/k)i,j is the reactivity introduced by the i-th isotope into the j-th zone and png,j and png,j are the
relative concentrations of fission products' in the fuel. Since the isotopic composition of the plutonium loaded
into the active zones of the reactor is identical, we obtain the following coupling equation:
``lip?, j*, (6)
in which j * is the number of the active zone with respect to which the concentration in any i-th zone of the
reactor is determined (Aj * = 1; A~ = 0 in the shields; Ai 0 0 in all subzones of the active zone).
By substituting Eqs. (2) and (6) into (1), (3), and (5), we obtain a nonlinear algebraic system of six equa-
tions with six unknowns which is solved as follows. First one does a neutron physics calculation of the base
version of the reactor in order to determine the cross sections, (6k /k), etc. Then one arbitrarily specifies
the values Tj in each zone and solves the system of equations o
(1), (3), and (5) in the six unknowns B and pi~J?
This system has siz solutions. However, only one of them satisfies the physical sense of the problem. Next
one determines pi and pi,j, and also the concentration of fission products and plutonium accumulation, com-
pares them with the specified values, corrects the values Tj, and repeats the procedure.. When necessary,
one calculates a new the neutron physics of the base version.
Calculations have been performed according to the proposed procedure of the physical characteristics
of a reference reactor [3]. A multigroup neutron physics calculation has been done according to a one-dimen-
sional program [4] using the BNAB-70 system of constants. The method for calculating (6k/k)i,j is taken from
[5]. The main expenditures of machine time in calculating the different versions are necessary for the neutron
physics calculation of the reactor.
The initial data necessary to determine the physical characteristics of the reactor, which were deter-
mined from the following formulas:
J 242
I4j
1'i [(1-Ej) P,j -Poi, i)]
IAi
Png,j
J 242
'Vi [N'- Ncf)j
j=1 i=239
EBR2 = J 242
Z (Nf)j
j=1 i=238
3 J
where yi = 71 Yi,jPi'j?j/ P~I,j?i is the relative cost of the i-th plutonium
=1 j=1
(Sk 239,j - (k)238,jNeiJ and Nj,j are the number of captures and fission of the i-th isotope in the j-th
zone; and EBR is the excess breeding ratio, are given in Table 1.
The reactivity loss upon production of a unit of energy is
J 242
1 dk i - 6k 6k
k a~ - T1 k (P.ii) +( k 1
where a is the heating power of the fuel.
The doubling time T2 = ln2/w was determined from the equation [1]
where Tnj is the average processing time of fuel from the j-th zone.
The annual excess amount of plutonium was calculated from the formula
7' 242 J J 242
?i ``~ EBR1 Ni P (B-1) `, ?i J o
r W 1 Tj !' Vi[(1-ej)p,,jPi[ = W ~. f. ng,i= W 1 T. yip
j=1 i-239 i-i j=1 i=239
k \ Sk
isotope [Ti,j=( S$ li.j - ( k )238,j/
J
\1 Fri
L~ y Png, i,
1-exp(-wTi/W) -~ 1--exp(-wTj/W) expt ( tp ~-7nj)J'
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where W is the reactor power. The specific load into the fuel cycle was found from the equation
J 242 242
G= W > (?i 7iPi+wP T; TiP?)
9=1 i=239 i=239
with c = 1, cj = 0, and Tnj = 1 year:
EBR1 .......................0.463
EBR2 ............. .........0.460
(1/k)(dk/d E), (MW ?day) -1 ... ...... ^ (-0.0002')
T2, years .................... 7.7
r, kg/(yr ? MW) ................ 0.178
G, kg/MW ..... . . . .......... 1.893
B .................. .....1.329
The dependence of r, (EBR1 + 1), B, G, and T2 on the depth of fuel depletion in the active zone is shown in
Fig. 1. These characteristics depend also to a significant extent on the rate of plutonium accumulation in the
shield and the losses upon chemical reprocessing, especially in the case of low depletion.
Steady-State Mode. In this case one writes the right-hand side of Eqs. (11) in the form Biki v (?;/
T;)PL39,; , where the coefficients ki=p!/p239 characterize the composition of the loaded plutonium. For the
steady-state mode one solves Eqs. (3) and (4) independently of Eqs. (1) with respect to only the two unknowns
P233,1 and p239,;* , determines all the remaining concentrations, and corrects the fuel delay times in the
zones, after which the operation is repeated. The dependence of EBR1 on the composition of the plutonium con-
sumed is shown in Fig. 2. As is evident from Fig. 2, an increase in k240 and a decrease in k241 result in an
appreciable increase in the integrated breeding ratio.
We note in conclusion that one can take approximate account of compensating elements within the frame-
work of the proposed method if one takes as the base version of the reactor one having an operational reserve
of reactivity - ( d~) E'' , where Ep is the specified energy production between regular rechargings.
Conclusions. An effective procedure is proposed for calculation of the loading and discharge rates of
fuel as well as its isotopic composition (in the continuous recharging mode), which is necessary for calcula-
tion of the physical characteristics of a reactor in the steady-state and natural fuel modes.
The procedure discussed permits determining the physical characteristics of the reactor in the natural:
mode as well as investigating their dependence on the composition of the fuel consumed.
The proposed algorithm makes possible the calculation of the physical characteristics with an arbitrary
dependence of the microcross sections, fluxes, enrichment function, and other parameters over reactor volume
and the optimization of reactors with a different heterogeneous structure.
The effectiveness of the procedure is achieved owing to the possibility of reducing a distributed problem
(with an arbitrary number of partition zones) to a point one (with one zone).
LITERATURE CITED
1. V. S. Kagramanyan et al., At. Energ., 46, No. 4, 236 (1979).
2. G. B. Usynin, At. Energ., 25, No. 6, 466 (1968).
3. A. Baker et al., in: Proc. Symp. on Calculation for a Large Fast Reactor, TRG Rep. 2133 (R), Risley
(1971).
4. A. I. Novozhilov et al., Kernenergie, 11, 329 (1975).
5. V. V. Orlov et al., ibid., 5, 112 (1969).
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i
1. Ya. Emel'yanov, V. V. Postnikov, UDC 621.039.562
and Yu. I. Volod'ko
The RBMK reactors have involved various difficult problems in control of core processes, particularly
in the monitoring and control of the power distribution. Reliable power-distribution control for reactors of
this type is complicated by various factors: the high unit power and large size of the reactor, the xenon in-
stability occurring in such reactors, the large number of monitoring points and control devices, which con-
stitute a considerable load of the operator, the complicated microstructure of the power distribution, which
is due to the large number of additional absorbers in the initial working period,- and to the large numbers of
adjacent fresh and partly worn out fuel-pin assemblies in the steady-state, in addition to the spatial instability
due to the void, temperature, and other reactivity coefficients [1],' and the complexity in monitoring the power
distribution by heat-engineering methods in channels with a boiling coolant.
Some of these factors become more important to the structure of the control system for a reactor of any
type as the size and unit power increase or as the specifications'for automatic control are tightened.
The monitoring and control system for the RBMK-1000 includes three basic systems each having highly
independent operation: the control and protection system CPS, the system for physical monitoring of the power
distribution CPMPD and the Skala centralized monitoring system CMS. We consider the main functions and
features of each system, with particular attention to the noval features of the structure and novel features of
the structure and novel items of equipment.
The CPS [2] operates from lateral ionization chambers LIC. In the upgraded form, the CPS is supple-
mented with local automatic control LAC subsystems and local emergency protection LEP systems, in which
triaxial fission chambers are used as the detectors [3],
The CPS is based to a certain extent on designs traditional for large uranium-graphite reactors and
provides for monitoring of the power and reactor period together with automatic maintenance of the reactor
power in the range 0.1-100%, as well as automatic emergency protection if the power level or rate of power
increase exceed set limits, automatic emergency reactor protection from deviations in the power distribution
at the periphery of the core corresponding to the first radial-azimuthal harmonics, monitoring of the relative
power distribution at the periphery of the reactor, and manual control of the positions of the absorbing rods.
The lateral ionization chambers are placed in the reflector (four chambers) and behind the reflector
(24 chambers), while there are 42 chambers within the reactor, whose positions are shown in the figure. The
effectors in the control and protection system are absorbing rods of three types, which are placed in 179
special zirconium channels and are cooled by water, including the manual control rods MC, which number 146,
the automatic control rods AC for the average reactor power, which number 12, and the short absorbing rods
SAR, which number 21. Fifty-seven rods in the manual control system are used as emergency-protection rods
and are withdrawn from the core during operation. The height distribution of the power is controlled by with-
drawing the MC and AC rods upwards, while the SAR' rods, which have absorbing parts of half the length, are
withdrawn downwards.
The LAC system is designed for automatic control of reactor power and to stabilize the power distribu-
tion [4]. When the reactor works with LAC, one of the AC must be in the hot backup state. Switching from LAC
to AC or vice versa is performed manually, or automatically, by means of the sevel effector organs for the
LAC, which are analogous to the MC effectors, and the 42 LAC and LEP chambers are uniformly distributed
over the core (Fig. 1). Each LAC rod is surrounded by two LEP chambers and four LAC chambers. The
averaged corrected signal from the four LAC chambers is used to control the rod.
The SPMPD consists of two independent subsystems for monitoring the power distribution over the
radius (SPMPDR) and the height (SPMPDH). Both systems work from the internal detectors ID that monitor
Translated from Atomnaya Energiya, Vol. 48, No. 6, pp...360-365, June, 1980. Original article submitted
May 14, 1979.
360 0038-531X/80/4806-0360$07.50?1980 Plenum Publishing Corporation
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to
AC -2
.... .. .. .. .. .. .. .. .
2
iL
23
,~ 22
10
15
14
W I WA I I LWJ !-I U U LW_I 1 1 !-Lj-! 1-1 !_! !_! 1_Lj_1 1 1 1-i't
L V''
A1 AC-2
14
0
AC -1
I Al CB-1
3
7 50 51 J63541jjjjllljff*I61
022
AC-2
(318 m9 010 011
16
0
AC-2
17
0
AC-3
0
AC+1
O20
Fig. 1. Location of the control rods and ionization chambers of the CPS and of the
SPMPD detectors in the RBMK-1000: 1) DMER; 2) channel for calibration y chamber;
3) detector for the LAC and LEP systems; 4) LAC and LEP rods; 5) AC rod; 6) MC
rod; 7) emergency-protection rod; 8) short absorbing rod; 9) DMEH; 10) fission chamber;
11) lateral ionization chamber (outside a reflector).
the radial power distribution DMER and the height distribution DMEH, which provide preprocessing of the
signals, transmission to the computer, comparison of the signals with given levels, and generation of auidable
and visible signals if the detector signals go outside set limits. The SPMPDR receives the signals from 130
DMER chambers placed in the fuel-pin assemblies, while the SPMPDH receives signals from 84 DMEH chambers
placed at sevel points along the height in 12 of the DMEH channels.
The number and location of the ID were chosen on the basis that the permissible power in a fuel-pin
assembly should only be slightly dependent on the power distribution over the height of a channel (up to axial
nonuniformity coefficients of 1.7-2.0) and is determined in the main by the flow rate, pressure, and tempera-
ture of the water at the inlet to the pin assembly. Therefore, the SPMPDR provides the basic internal monitor-
ing in the RBMK-1000; the DMEH chambers are intended in the main for monitoring the stability of the height
distribution and to prevent excessive linear load on the fuel-pin assemblies in anomalous situations. The
number and disposition of the DMER chambers are determined by the required monitoring and control accuracy.
The mean-square error in discrete monitoring of fuel-pin power at the maximum distance from the DMER
chambers is about 2% for the pitch chosen for the DMER lattice, which is much less than the errors required
to produce an appreciable deterioration in the determination of the margin from the critical power in the fuel-
rod assembly. Therefore, the_DMER lattice has some redundancy as regards the necessary monitoring ac-
curacy. The design is also intended to protect the fuel-pin assemblies from erroneous extraction of any in-
dividual automatic-control rod, where each control rod should be operated with at least one internal detector.
There are seven DMEH chambers uniformly distributed over the height because of the specification for monitor-
ing the first four axial harmonics even when two detectors in one channel fail.
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The DMER detectors resemble the LAC and LEP detectors in lying in the central dry sleeves of the
fuel-pin assembly, each of which is of the same design; a DMER chamber consists of a sensitive element
filled with argon in a sealed body of corrosion-resistant steel together with a sealed high-temperature socket
and a coupling line. The maximum working temperature of the sensing element in the DMER system is 350?C;
an emission detector with a silver emitter is used. This detector is a radiation-resistant and heat-resistant
coaxial cable made by a normal manufacturing technique used for high-temperature cables with mineral in-
sulation.
During the design of the SPMPD for the RBMK-1000, the choice of sensing element for the power-distribu-
tion monitors was essentially restricted to emission detectors with emitters composed of rhodium and silver.
Such detectors with silver emitters are rather faster in response [5], which is due to the shorter half-life of the
silver radioisotopes, as well as to more favorable 6-particle energy, which means that there is only a 13%
contribution or less to the detector current from the component with half-life of 2.4 min, while there is prac-
tically no contribution from the component with half-life 259 days. Also, a silver emitter has a lower rate of
burn-up (19% per year as against 34% per year for rhodium for a neutron flux density of 1014 neutrons/cm2
sec), together with a much lower dependence of the sensitivity on the neutron gas temperature. Further, a
cable with a silver core is easier to manufacture and more reliable, which is extremely important for routine
production and operation.
The working life of the DMER is not less than the working life of a fuel-pin assembly (3-4 years). Operat-
ing results led to some minor changes in the detector, which amounts mainly to simplification and replacement
of some of the soldered joints by welded ones. It was found that the period of a fault-free operation for such
a detector working with air in the central tube of the fuel-pin assembly at about 300?C (with the body removed)
was much less than that in argon, which is the filling used for the standard DMER, since the working life was
then only one year.
The seven DMEH are located in a dry sealed sleeve filled with a mixture of argon and helium. The
sleeve is set up in a channel analogous to the CPS channels and is cooled with water having an outlet tempera-
ture of up to 75?C. The sensitive element in a DMEH is the same as in the DMER and is a cylindrical spiral
of length 2.6 m. Periodic checks are provided by a tube having an internal cavity isolated from the volume
of the sleeve placed along the axis. The DMER and DMEH can be replaced, along with the LAC and LEP
chambers, with the reactor working. The maximum currents in the DMER and DMEH at the normal reactor
power are about 15 MA.
It is necessary to maintain a reasonably high insulation resistance Rin in the detectors, since Rin has a
direct effect on the current formation in the detector and thus on the accuracy [6]. The insulation resistance
varies only slightly during the working life and constitutes 10$-1010 0 with the reactor shut down for 90-95%
of the detectors or 107-1080 with a reactor power of 25-100% of nominal. If Rin falls below 5.105 S2, there
is a spontaneous change in the sensitivity of more than 10% that is unrelated to the input impedance of the
secondary apparatus, which is less than 100 52. The DMER and DMEH are checked mainly by scanning the
fuel-pin assemblies adjacent to the DMER as well as the central sleeves of the DMEH with the reactor working
by means of small triaxial fission chambers.
The secondary electronic equipment in the SPMPDR differs from the apparatus in other such systems
mainly in that it combines the functions of monitoring the power distribution and producing signals when
permitted levels are exceeded with the function of monitoring the relative power distribution, which includes
outputting information on deviations from the specified distribution. The relative power distribution is mon-
itored by comparing the reference levels and corresponding signals for each of the DMER as normalized to
the total signal from the DMER. A display in front of the operator presents signals when the reference levels
are exceeded by more than 5 and 10% and when the levels fall by more than 10%. The normalization reduces the
error of the system arising from the lag in the DMER during fast changes in reactor power, while the informa-
tion displayed to the operator on the form of the power distribution is unaltered as the power changes. The
operator can estimate the deviations from the specified relative power distribution by altering the reference
levels in the relative monitoring channel over the range f 15%. The total-current recorder for the DMER is
the main instrument for monitoring the reactor power.
The secondary electronic equipment in the SPMPDH also combines the functions of absolute and relative
monitoring of the power distribution. The structure of this system differs from that of the previous one only
in that the signal from each DMEH is normalized to the total current of the seven DMEH in one channel, not to
the total current of all detectors.
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The SPMPD provides monitoring of the power distribution in the range from 10 to 100% of the nominal
power as well as monitoring of the thermal power of the reactor in the range from 5 to 100%.
The Skala central monitoring system [7] is based on a V-3M computer and provides monitoring, process-
ing, display, and recording of most of the engineering parameters of the reactor and of the power station as
a whole. The signals from the central system are processed by the PRIZMA program, which calculates the
power for all the fuel-pin assemblies, the limiting permissible power for each fuel-pin assembly correspond-
ing to a given probability of crisis-free operation in the assembly, the reserve-power coefficient Kr relative
to the limiting acceptable power for each fuel-pin assembly and the same for the reactor as a whole, calcula-
tion of the steam content in each fuel-pin assembly, the power production by each assembly and by the reactor
as a whole, the reactivity margins, the burn-up in the DMER and DMEH, the recommended water flow rates
through the fuel-pin assemblies, the reference and maximal signal levels for the DMER and DMEH, the am-
plitudes of the axial harmonics representing height deviations in the energy distribution at the points bearing
the DMEH assemblies, overall reactor parameters (reactor power indicated by the SPMPD and the heat-
engineering instruments, nonuniformity coefficients for the power distribution, total power and flow rate in
individual parts of the reactor), the recommended displacements for the control and protection system rods,
etc. PRIZMA also performs the diagnostic processing for certain forms of reactor equipment and also ac-
cumulates data on the peripherals for subsequent statistical analysis, while providing cyclic recording of data
for analysis of emergency situations. Any cell in which the calculated or measured parameter (Kr, fuel-pin
assembly power, water flow rate, etc.) differs from the specified one is displayed to the operator. The results
are printed out as patterns with indication of the type of each channel (fuel-pin assembly, MC, SAR, AC, DMER,
etc.) together with the parameters channel by channel: water flow rate, steam content, position of CPS rod.
The pattern is accompanied by a brief summary of the general reactor parameters and a list of the 6-10 most
highly stressed fuel-pin assemblies with the maximum power and minimum Kr. The calculations are per-
formed in 5-8 min.
The software for the operation of the RBMK is designed to overcome the complexity or impossibility of
direct monitoring of many parameters, including ones related to safe operation of the core. These parameters
are monitored indirectly by calculations in the Skala system on the basis of results from more complex cal-
culations performed by an external BESM-6 computer. The data exchange between the BESM-6 and the central
monitoring system for the RBMK-1000 at Kursk and Chernobyl nuclear power stations is performed automatic-
ally by means of Akkord 1200M interfaces. The software for operating the RBMK-1000 includes the following:
PRIZMA for the in-station computer, BOKRUS for periodic physical calculations on the BESM-6 giving the
opimum power distribution and the positions of the control rods, BOKR, LEN, and KVARTs for BESM-6 pro-
cessing of the measurements involved in checking the DMER, and ANALOG for periodic checking of the
PRIZMA with the BESM-6.
The profiles for the distributions of parameters such as the fuel-pin power, the channel water flow rates,
and the reference levels in the SPMPD substantially influence the safety and economy of the RBMK. A param-
eter that is to be used in optimizing any of the parameters of the reactor in design and operation is obviously
the referred cost p per kWh subject to the various constraints imposed primarily by the requirements of
power station safety. In individual instances where most of the parameters have been specified, it is permis-
sible to optimize a single parameter that influences 17. For example, in defining the mutual distributions of
the power distribution and flow rates, one uses the maximum probability of crisis-free operation in the core
for the RBMK-1000. The basic input data are here the specified macroscopic distribution for the power pro-
duction over the radius, which influences the limiting permissible reactor power, and the specific power pro-
duction in the fuel, together with the dynamic stability in the power distribution and the reactivity balance. The
specified macroscopic power distribution is determined from optimization calculations on the external com-
puter for each enlarged state of operation.
The sequence for controlling the power distribution in the RBMK-1000 includes observation on the power
distribution from the SPMPD signals, the display, and the printout patterns from the special monitor, together
with the parameters related to the power distribution (distribution of Kr, distribution of the water flow rates,
and general loop engineering parameters), which may involve outputting recommendations on changing the water
flow rates, the reference levels in the SPMPD, the positions of the correctors for the CPS chambers, all of
which employ recommendations from the central monitoring system. Further, there may be actions involving
direct control of the power distribution by displacement of the control rods and by altering the levels of the
AC or LAC transducers, together with changes in the modes of operation of the monitoring and control systems
(switching of the AC and LAC, calls for recording SPMPD signals, the calling of central monitor system data,
etc.). The number of manual operations involving the CPS rod is reduced in stationary state by factors of
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2-5 by transforming from automatic control to local automatic control. If power overload occurs in a fuel-pin
assembly, the power distribution can be regulated with or without the LAC, and the work of the operator is
simplified in the first case. Each reloading operation for a fuel-pin assembly is preceded by adjustment of
the flow rate in the reloading region. Manual control of the power distribution is provided by alternate opera-
tion of the manual controls and the SAR, which eliminates general reactor or local deviations in radius and
height.
The monitoring and control systems thus constitute a hierarchic structure with virtually independent
base subsystems, which are combined as regards monitoring functions by means of the central monitoring
system, which lies at the top. Such structures provide high reliability and viability in the entire system, but
they impose a considerable load on the operator and other staff, who are involved in ensuring consistent opera-
tion of the individual low-level systems via manual control of the power distribution and so on.
It has become necessary to improve the control systems for the power distribution in the RBMK reactors,
and the following factors are involved:
1) increasing the specific thermal loads on the fuel and other parts of the core, with substantial increase
in the number of monitoring points and number of control units for the next generation of RBMK (RBMK-1500
and RBMKP-2400);
2) the ongoing tightening of specifications for reactor safety and reliability in nuclear power station
3) the need to reduce the load on the operator and to improve the economy of the power station; and
4) the need for the power station to operate in load-tracking modes.
Experience with the RBMK-1000 has given rise to practical proposals for improving these reactors; in
order to improve the power-distribution control systems, it will be necessary to make further use of com-
puters in the light of the above factors. Extension of the sphere of the computers in control involves new
problems, particularly safety aspects. It is necessary to define the correct combination of analog independent
systems with systems based on standard computers. One possible approach is a combined monitoring and
control system that includes the CPS traditional for the RBMK-1000 that works with the lateral ionization
chambers, together with a system for monitoring and controlling the power distribution (SMPD) employing
analog instruments working with the 252 radial and 80 height internal detectors and providing local protection
against major deviations from the specified margins for avoiding crisis and linear heat loading in the fuel-
pin assemblies, together with transmission of the normalized signals from the internal detectors to the
analog local automatic controls and computer, and finally a control computing system consisting of several
subsystems each with its own computer and a higher-level computer, or else one in which each subsystem
employs several computers.
Such a system will have a hierarchic, structure, and the first level contains analog systems (CPS and
SMPD), which normalize and transmit detector signals to the second-level systems, which employ computers
and which process the data to present information to the operator while providing automatic management of
the energy distribution (via the CPS), as well as diagnosis of equipment states and local emergency protection
on the basis of current crisis margins and heat loads on the fuel-pin assemblies. At the third level there is
a high-power computer that performs complicated physical and optimization calculations and which provides
coupling to the external computer in the power system.
The independent digital protection subsystem in the second level calculates the maximum permissible
signals from the internal detectors and compares these with the current signals, which provides a digital
protection function, while the CPS and SMPD provide analog protection. The control functions of the CPS and
SMPD back up the analog functions executed by the control computing system. The DMER are provided by
fast emission detectors of cable type, in which the emitter contains hafnium, while the DMEH are assemblies
of two triaxial gamma chambers. Each of the 20 channels used in height monitoring employs dry sleeves
cooled by water in the CPS loop, and four such assemblies are set up in each channel. A characteristic fea-
ture of the system is that it provides local automatic emergency protection of the reactor from erroneous
removal of any control rod. The system is designed for completely automatic control of the power distribu-
tion under stationary and transient conditions.
Future improvement of the automatic facilities and computer backup for power stations with RBMK
should not be accompanied by an increase in the amount of information output to the operator, but instead
should involve automating control and output of information to the operator only of essential data indicating
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anomalous situations not envisaged in the monitoring and control algorithms. The operator should be protected
from redundant routine information in order to be able to concentrate attention on the basic factors that deter-
mine the safety and efficiency of the reactor.
1. A. P. Aleksandrov and N. A. Dollezhal', At. Energ., 43, No. 5, 337 (1977).
2. I. Ya. Emel'yanov, in: Nuclear Science and Engineering: Reactor Physics and Engineering Series,
Issue 1(21) [in Russian], TsNllatominform, Moscow (1978), p. 36.
3. I. Ya. Emel'yanov et al., At. Energ., 43, No. 1, 44 (1977).
4. I. Ya. Emel'yanov et al., in: Nuclear Science and Engineering: Reactor Physics and Engineering.Series,
Issue 1(5) [in Russian], TsNllatomihform, Moscow (1979), p. 3.
5. I. Ya. Emel'yanov et al., At. Energ., 34, No. 3, 203 (1973).
6. I. Ya. Emel'yanov et al., ibid., 37, No. 1, 72 (1974).
7. V. I. Adas'ko et al., "Systems for monitoring and controlling engineering processes in nuclear power
stations by means of control computers," International Electrotechnical Congress, Moscow, June 1977,
Section 7 [in Russian].
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V. N. Smolin, S. V. Shpanskii,
V. I. Esikov, T. K. Sedova,
and V. P. Shishov
Thermohydraulic stability of the operating conditions in parallel steam-generating channels is one of
the factors which ensures the reliability of cooling of the fuel elements of nuclear reactors. The problem of
determining the limit of thermohydraulic stability was first formulated and solved at the contemporary level
in the research of Morozov [1]. The stability of a one-dimensional time-dependent model of flow in a channel
incorporating the linearized equations of thermohydraulics with empirical relations closing them was invest-
igated in his research. Some theoretical questions associated with the application of such an approach to com-
plex systems have been discussed by Molochnikov [2].
A program for calculating the limit of thermohydraulic stability by the D-partition method was proposed
in [3] which is based on the algorithm of [1]. However, the program of [3] permits calculating only a vertical
channel of constant cross section without local drag in intermediate cross sections. The evaporative channels
of boiling reactors constitute a complex system of sections of various transfer cross sections. They can in-
clude sections with various slopes and have local drag along the channel length. Therefore, the problem of
practical implementation of the method of calculating stability in flexible and efficiently running computer
programs has still not been definitely solved.
We will discuss the system of equations of the thermohydraulics of a one-dimensional fluid flow which
includes the continuity, momentum, and energy equations, which have the following form for a channel of con-
stant cross section without local drag (without the kinetic energy and pressure energy taken into account):
OP + aG 0'
OT W =
+ f az (GIv) -I- I ap = - gpf sin 0 -
2-1- f ti + az (Gi) = lI?q? lIigl.
It is taken into consideration here that the wetted perimeter of the channel may consist in the general
case of two surfaces - an "outer" one (the surface of the channel lining) and an "inner" one (the sur-
face of the rods).
For a steady flow, system (1) takes the form:
G (z) = G (0);
rZ
t (z) = i (0) + ~ o J (fl?9 -I- n ) dz;
0
z
P (z)=P (0)+ j [v(z)-v(0)]+gsin0 pdz+
0
+ fo J (II?T?+IIiT1) dz.
0
After linearization, a Laplace transformation, and reduction to canonical form, system (1) for small
deviations (variations) of the operating parameters is converted into a system of ordinary differential equa-
tions whose vector form is
Translated from Atomnaya Energiya, Vol. 48, No. 6, pp. 366-369, June, 1980. Original article submitted
July 23, 1979.
0038-531X/80/4806-0366$07.50 ?1980 Plenum Publishing Corporation
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dX/dz = AX, (3)
where X is a vector whose components are the Laplace-transformed variations of the flow rate, pressure, and
enthalpy:
1G (s, z)
X(s, z) = p (s, z)
a (s, z)
i (s, 0)
P = p (s, 0)
1)
is discussed for system (3).
Any solution of system (3) has the form
X (s, z) = ct (s, z) X (s, 0),
where 4)(s, z) is the fundamental matrix of system (3) normalized at z = 0, whose columns are linearly indepen-
dent solutions of the system (3):
Gt (s, z)
G2 (s,
G3 (S, z)
cli (.s, z) = ~Pt (s, z)
Pz (s,
P3 (S, z)
Lt (S, Z)
l2 (S,
L3 (S, Z)
F (s) = i Pt (s, H) P2 (s, H) P3 (s, 17)
Using Eqs. (5) and (6), we obtain an equation which relates the solution of system (3) to the vector of the
boundary conditions, viz.,
F (s) cP-1 (s, z) X (s, z) = F.
By substituting Eq. (7) into Eq. (8) and expanding the determinant, we obtain the characteristic equation
Thus, there is no need to find the fundamental matrix in order to analyze the stability of system (3), but
it is sufficient to find a solution with the initial conditions: G(s, 0) = 1, p(s, 0) = 0, and i(s, 0) = 0.
This conclusion is valid for a channel with local drag along its length or with variation in its cross
section. In this case the characteristic equation is of form (9). The channel is treated as a series of smooth .
sections with constant cross section whose boundaries are cross sections with local drag or variation or cross
section. The splicing of the solutions at the boundary of sections is accomplished on the basis of a well-known
relation for the pressure losses by local drag, which is considered as quasisteady, with the introduction of a
term which takes account of the pressure difference due to variation of cross section:
OP (za) _ d t2 (za+0) -1) 2J2 ~Td _O) .
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'6P,
MPa
0,25
Fig. 1. Comparison of the experimental and calculated thermohydraulic characteristics of a model
of the circuit of a channel of the RBMK-1500 reactor (a) at a pressure of 4.8 MPa and a power of
265 kW and (b) at a pressure of 6.87 MPa and a power of 614 kW: e) circuit, O) channel, A) zone
of heat generation, p) exit section of the channel, 0) steam-water communication, and ?) water
communication.
Fig. 2. Comparison of experimental data with the calculated stability limit at a pressure of 3.0
MPa and (a) a power of 260 kW and (b) 615 kW: 0) pulsation-free conditions, *),region of pulsation
conditions.
c,
kg/sec
2,5
4O
0,5 L
160
Fig. 3. Comparison of the experimental data with the calculated stability limit at a pressure of
5.0 MPa and (a) a power of 265 kW, (b) 620 kW, and (c) 1000 kW. The symbols are the same as
in Fig. 2.
Fig. 4. Comparison of experimental data with the calculated stability limit at a pressure of 7.0
MPa and (a) a power of 260 kW, (b) 625 kW, and (c) 1015 kW. The symbols are the same as in
Fig. 2.
It is assumed that the flow rate and the enthalpy do not vary due to local drag. These conditions lead
after linearization and Laplace transformation to the splicing equation in cross section zd:
X(s, zd -{- 0) = Fd X (s, zd - 0),
where the matrix Fd has the form
1 0 0
Fd = (Kc 1 Kt ) ;
0 0 1 ,
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KG G 12( zd-0-1+~ ) 2v4-G a l
-212(zd-0) /2(zd+0) d / dG l'
G2 au /2 (zd -0) 11
Kt 2/2(zd-0) a1 \/2(zd+o)+`d)
In order to close this system of thermohydraulic equations, a number of empirical relations are used .
which are considered as quasisteady. The hydraulic friction is taken into account on the basis of the formula
APg =Kam~ TG
,
h
h
T?II?-)-TL111=Kam g-'z (??II?+X1H').
The coefficient of hydraulic friction of a single-phase flow is determined from the formula [4]
x=0.1 (1.46Aj 100 Re)0.25.
The coefficient which takes account of the difference between the friction of a two-phase flow and that
of a single-phase flow is found from the equation
Kam=l+3cp(1-(p2)( ~4 , 221 P .15 -1).
The form of the relation is obtained by analysis of the data of [5], and the numerical coefficient "3" is
selected according to the results of a determination of the hydraulic characteristics of a model of an RBMK-
1500 channel [6].
4 = qhg bwCwpw ddti ,
where qhg is the thermal flux due to heat generation which is constant in time and bv, is the geometrical char-
acteristic of the channel wall (the ratio of the area of the transverse cross section of the wall to the perimeter
of the heat exchange). The wall temperature for a single-phase flow is calculated with the help of the formula
The procedure has been adopted for calculating the steam content of a two-phase flow which is proposed
in [7] and modified in [8], which includes the equation given below for the start of boiling, shear, and enthalpy
of the liquid phase. The enthalpy of the start of intense surface boiling has been adopted as the limit of a two-
phase flow viz.,
i -i'-7.5 q1 ( gdh 10.0s 4G )0.2.
sb G 1 rp"v, / (n?
We determine shear of the phases from the formula
(1.062+2.G54/(1+ 1 x u\2) dh2s
s+ X. V J J .. n
2
~G/1cl \ 21.15
The enthalpy of the liquid phase is calculated from the equation
=i'-(i+i'-2isb)e
We find the true mass and volume steam content from the formulas
L-'Z
x=
i -' l
LP=1 1~ 1-xs_v
The roots of Eq. (9) are analyzed. by the D-partition method; the boundary of the regions of existence of
different numbers of roots with a positive real part is found by specifying the purely imaginary values of the
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Laplace transformation parameter s = jw and it is checked whether or not the operating point belongs to a
region which does not contain a single root with a positive real part.
As has been shown in [11, it is possible in the majority of practical cases to neglect the effect of a pres-
sure variation on the remaining parameters. Then the continuity and energy equations in the system (3) are
independent of the momentum equation, and after transformation for purely imaginary values s = jw they take
the form:
dGR
dz
dG7
dz= A1GR - A2LR;
jZ = A3GR +A4GI + A5iE + A6II;
z
_ - A4GR -r A3GI - AIIR A5L7
integrated momentum equation for a smooth section of the channel takes the form
tend Zend
APR = - f J GI dz + (A7GR A5iR) dz +
7st Zst
B1 (zend) GR (zend) - B1 (z st ) GR (z st ) -I- B2 (zend) LR (zend) - B2 (zst
z end zend
r
API= ril f GRdz+ I (A1G,+Asi,) dz+
ZSt Zst
IR (ZSt
+ B1 (zend` GI (zend) - B1 (z St ) GI (zst ) + B2 (zend) i, (zend) - B2 (z St ) 'I (z sty )
APR = zG (27 + Ro 3 Ae GR (0);
Azo
API = f GR (0)
The complex plane pt (s, H), on which the D-partition curve pt (j w, H) = pR(w, H) + jpl(w, H) is constructed,
is selected as the D-partition plane. The values of PR(w, H) and pl(w, H) are determined from Eqs. (11) and
(12), in which the solutions of the system (10) with the initial conditions (13) are used. The "operating point"
on the D-partition plane is the origin of coordinates PR = 0 and pl = 0. Thus, the system is stable if the curve
pj(jw, H) intersects the PR axis in the positive region.
The outlined procedure for calculating stability has been implemented in the computer program CAHS
(calculation of hydrodynamic stability). The program is written in ALGOL for a BESM-6 computer and permits
calculating the limit of both aperiodic and oscillatory instability of a steam-generating channel consisting of
sections of different geometry and having local drag in arbitrary cross sections. The program CAHS has been
tested by comparing the computational results from the program with experimental data on stability limits in
a model of the circuit of a channel of the RBMK-1500 reactor [6].
The model of the circuit consisted of water communication at the input into a channel of length 11.4 m,
the heat-generation zone of the channel consisting of a bundle of seven rods with an outer diameter of 13.5 mm
and a length of 7 m, the output part of the channel -11 m in length, and steam-water communication at the
channel exit -29 m in length. The circuit was divided into 51 computational sections. All the spacer lattices
were taken into account as local drag. The heat capacity of the walls of the heat-generating rods and thermal
losses along the circuit were taken into account.
GR(0)=1
GI(0)=iR(0)-=iJ(0)=0
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Comparison of the experimental hydraulic characteristics of the circuit and its sections with those
calculated from Eq. (2) with the empirical dependences given above is presented in Fig. 1. Sufficiently satis-
factory agreement of calculation with experiment is obtained, which indicates the validity of these dependences.
The stability limits are calculated for pressures of 1.0, 3.0, 5.0, and 7.0 MPa and powers in the channel
of 90, 250, 600, and 1000 kW. Comparison of the results of the calculation with the experiments is given in Figs.
2-4. Good agreement for this kind of calculation is obtained with the experiments over the entire parameter
range investigated.
a - thermal conductivity, m2 /sec
Cp - specific heat, J/kg ? deg
dh - hydraulic diameter, m
f - transfer cross section, m2
g - gravitational acceleration,
G - coolant flow rate, kg/sec
H - total length of channel, m
i - enthalpy, J/kg
I - volume enthalpy, J/m3
j - imaginary unit
p - pressure, Pa
If - perimeter, m
r - heat of vaporization, J/kg
q - thermal flux, W/m2
m/s ec2
s - Laplace transformation parameter
T - tangential stress, N/m2
v - specific volume, m3/kg
x - mass steam content
z - coordinate along the channel axis, m
a - heat-exchange coefficient, W/m2 ? deg
A - absolute roughness, m
cp - true volume steam content
At - thermal conductivity, J/m ? deg
? - dynamic viscosity, Pa
v - kinematic viscosity, m2/sec
p - density, kg/m3
0 - angle of inclination to horizontal
T - time, sec
co - frequency, Hz
t - temperature, ?C
st, end - start and end of the channel section under discussion; ('), (") - water and steam on the satura-
tion line; 1 - liquid phase; d - local drag; o, i "outer" and "inner" parts of the perimeter; R, I - real and
imaginary parts of a complex quantity; and - - a quantity which has been Laplace-transformed.
1. I. I. Morozov and V. A. Gerliga, The Stability of Boiling Equipment [in Russian], Atomizdat, Moscow
(1969).
2. Yu. S. Molochnikov, in: Some Questions of the Reliability of Nuclear Reactors [in Russian], A. I. Klemin
and M. M. Strigulin, Atomizdat, Moscow (1969), p. 289.
3. M. S. Lavrova et al., Preprint IAE-2238, Moscow (1972).
4. I. E. Idel'chik, Handbook on Hydraulic Drag [in Russian], Energiya, Moscow (1975).
5. G. Wallace, One-Dimensional Two-Phase Flow [Russian translation], Mir, Moscow (1972).
6. V. N. Smolin et al., "Investigation of the thermohydraulic stability in a model of the circuit of a channel
of the RBMK-1500 reactor," Lecture at the Sixth All-Union Conference on Heat Exchange and Hydraulic
Drag in Units of Power Machines and Equipment, Leningrad (1979).
7. P. Kroeger and N. Zuber, Int. J. Heat Mass Transfer, 11, No. 2, 211 (1968).
8. V. S. Osmachkin and V. D. Borisov, Preprint IAE-1957, Moscow (1971).
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NEUTRON DISTRIBUTIONS
Neutron distributions in reactors are commonly calculated by the homogeneous-diffusion method in which
all cells of the reactor are replaced by homogeneous cells, and neutron transport in them is described by the
diffusion equation
DIM) (r)-Eac)(r)+S'=0. (1)
The coefficients D', E' (homogeneous constants) and Si are constant within a single cell. The traditional
method for obtaining Ea is to average the true absorption cross section over the neutron flux obtained from a
kinetic calculation of an infinite uniform lattice consisting of cells of the i-th kind. A homogeneous-diffusion
cell with Ea is equivalent to the initial cell with respect to the probability of neutron absorption in it. There
is no direct relation between Di and any characteristic of the initial cell. In this method of homogenization
an approximate relation between D and Etr is established by making the simplifying assumptions that either
1a 0.2; for m < 0.2 it is necessary to use intervals with a larger value of the time. We note that these state-
ments give concrete form to certain prognostic conclusions in [6] where the dependence of the slowing down
density on m was analyzed for a source E0 = 2.45 MeV. For the general character of the time, that for a
stratum penetrated by a hole, the epithermal neutron flux is cut off sharply at a certain t; the cutoff time
increases with decreasing m and with the displacement of the instrument from the axis of the hole.
Computation Time and Errors. The calculation of all the curves of Figs. 1-4 required 42 h of machine
time on a B SM-6. The standard relative error of the results was ^?10-15%.
With an increase in the probe distance z there is an increase in the volume of computational work
(v.c.w.) necessary to achieve a given accuracy. Thus, in the eccentric position for epithermal neutrons in the
time interval 10-7-10-5 sec the values of the v.c.w. to obtain the same accuracy of the results at probe dis-
tances z = 0, 10, 20, 30 cm will be approximately in the ratio 1 : 2: 7: 9. For thermal neutrons the estimate of
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the v.c.w. will evidently be the same. For fast neutrons the v.c.w. increases more rapidly with increasing
z. For example, for Eo = 1.4 MeV in the centric position in the energy range 100 eV-4.65 keV the values of
the v.c.w. for the neutron density close to the time maximum for probe distances z = 0, 10, 20 cm are in the
approximate ratio 1: 16: 400.
All the numerical values of the neutron fluxes and their statistical errors are given in a report of the
Computation Center of the Siberian Branch of the Academy of Sciences of the USSR.
'LITERATURE CITED
1. A. A. Morozov and A. I. Khisamutdinov, Preprint No. 78 VTs Sib. OtdAkad. ; Nauk SSSR, Novosibirsk
(1977).
2. Evaluated Nuclear Data Library, Lawrence Livermore Laboratory (1975).
3. L. P. Abagyan et al., Group Constants for Calculating Nuclear Reactors [in Russian], Atomizdat, Moscow
(1964).
4. R. A. Rezvanov, V. Ya. Gommershtadt, and V. E. Lebedev, in: Nuclear Geophysics, Proc. of VNII of
Nuclear Geophysics and Geochemistry [in Russian], No. 7, Nedra, Moscow (1969), p. 75.
5. Yu. S. Shimelevich et al., Physical Principles of Pulsed Neutron Methods of Studying Boreholes [in
Russian], Nedra, Moscow (1976).
6. R. A. Rezvanov, Fiz. Zemli, 3, 105 (1970).
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 Declassified and Approved For Release 2013/02/01: CIA-RDP10-02196R000800030005-9
DETERMINATION OF THE DETAILS OF THE RESONANCE
STRUCTURE OF BOTH THE TOTAL CROSS SECTION AND
THE FISSION CROSS SECTION OF 235U AND 239Pu FOR
2 eV-20-keV NEUTRONS
A.
A. Van'kov, Yu.
V.
Grigor'ev,
V.
F. Ukraintsev,
T.
Bakalov,
G.
Ilchev, S. Toshkov,
Chan-Khan'-Mai, and N. Yaneva
When fast reactors are calculated with the aid of the system of BNAB group constants [1], one must know
both the average group cross sections of the fuel nuclides and the design materials and the factors of their
resonance self-shielding. These factors are calculated from the resonance parameters of the corresponding
nuclides. In the range of unresolved resonances, the errors of both the computer model and the initial data
(average resonance parameters) imply considerable errors of the factors of self-shielding (^-20-30%) [2]. It
is therefore necessary to directly measure the average characteristics of the resonance structures of the
cross sections at resolved and unresolved resonances.
The present work presented the results of measurements of those characteristics; the measurements
were made with the technique of neutron transmission and self-indication in 235U and 239Pu: the average total
cross sections and the factors of the resonance self-shielding of the total cross section and the fission cross
section were determined at 2 eV-20 keV.
Experimental. Technique. The transmission function and the fission self-indication function
(T (x)) = J 'P (E) e (E) exp [ -Qt (E) x] dE' J y (E) e (E) dE; (1)
4E 4E
(Tf (x)) = J W (E) of (E) exp [ at (E) x1 dE' J tC (E) 6t (E) dE (2)
AE AE
were calculated, where at(E) and af(E) denote the total cross section and the fission cross section of the iso-
tope under consideration; c (E), neutron spectrum; e(E), efficiency of the neutron detector; x, sample thickness;
and DE, interval over which the energy was averaged.
The experimental (T(x)) and (Tf(x)) values can be used to determine the parameters of the resonance
structure of the cross sections and the factors of resonance self-shielding of both the total cross section and
the fission cross section [3]:
ft = ((,it) C \ \at-f-ao// > 1 (where (Tmax denotes
the cross section at the resonance maximum). Figure 1 shows the equipment-dependent spectrum of the
counters with 3He in the measurements involving a 0.00822-nuclei/b (1 b = 10-28 m2) plutonium sample and
manganese and tungsten filters. Figure 2 shows the equipment-dependent spectrum of the uranium chamber in
measurements with a 0.00515-nuclei/b uranium sample and the same filters. The energy range above 3 keV
falls within the first few dozen channels. In this region the background was determined by placing into the
beam a titanium filter which led to the development of resonance minima at 3.8 and 18 keV. The dependence
of the background upon the channel number was approximated with a polynomial on a computer and subtracted
from the instrument spectrum.
The background in the beam without sample reached 2-5% in the case of the detector with 3He and
amounted to 20-40% for the maximum of the sample thickness. In the case of the fission chambers, the cor-
responding figures were 5-15% in the beam without sample and up to 30-60% at the maximum of the sample
thickness.
Results of the Measurements. Figure 3 shows the thickness dependence of the transmissivity and the
fission self-indication of 235U and 239Pu for the energy intervals assumed in the system of the BNAB constants.
The error of the results depended basically upon the error made in evaluating the background measurements.
The statistical error of the transmissivity at the center of a group did not exceed 0.5% in all measurement
series.
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Some time ago [7, 8] the results of measurements of the self-indication of uranium and plutonium fis-
sion in the thickness ranges 0.00035-0.014 and 0.00015-0.0079 nuclei/b, respectively, were published. Our
data on the self-indication are in rather good agreement with those data within the thickness and energy
intervals compared.
In order to determine the parameters of the structure of the cross section in the form of subgroup
parameters [3], the transmissivity and self-indication functions were processed with programs of the least-
square method. These parameters were used to determine the factors of the resonance self-shielding for
the various cross sections ao of thinning. The data are indicated in Tables I and 2. Figures 4 and 5 show the
energy dependence of the factors of self-shielding of the total cross section ft(ao) and of the fission cross
section ff(o-o) for the cross sections 0.100 and 1000 b of thinning. The figures include for comparison the
calculated foreign data [9, 10] and the FEI determinations made on the basis of BNAB-78 (calculations of [1]
with certain corrections). The error of the factors ft of self-shielding at 1-21 keV and uo = 0 amounts to 8-
10%, but reaches only 5-7% below 1 keV. For the factors ff, the error amounted to ^-5% at the same energy to
3-5% at energies below 1 keV. The accuracy of ft and ff increases with increasing ao. The results for (at)
are listed in Table 3.
Conclusions. It follows from Fig. 4 and 5 that the spread in the calculations of the factors of resonance.
self-shielding is extremely large. Our data for 235U and 239Pu are in regard to the factors ft systematically
below the calculated values at energies above 1 keV. The experimental values are much larger than the values
calculated with BNAB for the factors ff of 239Pu.
The authors consider it a pleasant duty to express their gratitude to L. B. Pikel'ner, M. N. Nikolaev,
and A. M. Tsibul' for useful discussions.
1. L. P. Abagyan et al., Grouped Constants for the Calculation of Nuclear Reactors [in Russian], Atomizdat,
Moscow (1964).
2. Dermott E. Cullen and F. Ernest, Trans. Am. Nucl. Soc., 17, 490 (1973).
3. M. N. Nikolaev et al., At. Energ., 30, No. 5, 426 (1971).
4. B. Bemer, A. A. Van'kov, and Yu. V. Grigor'ev, Prib. Tekh. Eksp., 6, 57 (1974).
5. A. A. Bogdzel' et al., Communication of the Joint Institute of Nuclear Research 3-9012 [in Russian],
Dubna (1975).
6. A. A. Van'kov et al., in: Nucl. Data for Reactors, 1, 559, IAEA, Vienna (1970).
7. R. Bramblett and J. Czirr, Nucl. Sci. Eng., 35, No. 3, 350 (1969).
8. J. Czirr and R. Bramblett, Nucl. Sci. Eng., 28, No.1, 62 (1967).
9. R. Kidman and R. Schenter, Group Constants for Fast Reactor Calculations, HEDL-TME-71-36 (1971).
10. E. Menapace, M. Motta, and G. Panini, A 26-Group Library with Self-Shielding Factors for Fast
Reactor Calculations with the UK Nuclear Data Files, RT/FI (73) 15 (1973).
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 Declassified and Approved For Release 2013/02/01: CIA-RDP10-02196R000800030005-9
WAVEGUIDE PROBING USED FOR THE DOSIMETRY OF
BREMSSTRAHLUNG IN HIGH-CURRENT ACCELERATORS
Yu. P. Bakulin, A. P. Korotov Kikh, UDC 539.12.08
and N. N. Morozov
The development of experimental investigations in the bremsstrahlung fields of high-current accelerators
is closely related to the development of dosimetry techniques which must have broad dynamic ranges from 103
to 1010 A/kg and a resolution of nanoseconds. Most widely employed are techniques based on luminescent and
optical properties of solids [1-3] and semiconductor detectors [4] at dose rates below 107 A/kg [5-7].
In the last few years methods having an upper limit of the dynamic range at 1010 A/kg were proposed;
these methods are based on direct-charge detectors [8], pyroelectric detectors of radiation [9], and bolometers.
But the problem of developing methods with a metrological foundation and instruments for absolute measure-
ments of the dose rate of ionizing radiations with high intensity was not solved because of the well-known dif-
ficulties encountered in calibrating the instruments.
Calibration in reference fields with a low dose rate (10-2 A/kg) and with a monoenergetic radiation
spectrum, followed by the transfer of the units to the range of high dose rates, is most frequently employed.
This technique does not allow to create a measuring system with errors consistuent with the requirements of
practical applications.
The method: listed above are based on solid-state converters of the energy of the ionizing radiation,
though when work is done in a field of pulsed photon radiation, the use of a material of low density (gas) is not
less promising. It should be noted that the physical limit of the ionization technique is 1015 A/kg [10] but con-
ventional ionization chambers allow measurements of the exposure dose rate of at most 102 A/kg with a re-
solution not greater than 10-5 sec.
Superhigh quality probing makes it possible to extend the dynamic range of the ionization technique to
109 A/kg with a resolution of nanoseconds [11, 12]. For measuring a dose rate in excess of 103 A/kg with a
probing technique with a waveguide is most promising.
Absolute measurements of the dose rate with the aid of ionization techniques suggest that a volume of
homogeneous sensitivity be employed. In our work we used a wall chamber in the form of a cylindrical 5-mm-
thick radiator made from an air-equivalent plastic (C5H802)n with an air-filled Bragg plane (height -0.5 mm).
The measuring waveguide had the form of a strip line of special design the conductors of which were distributed
on flat surfaces enclosing a cavity. The conductors were in the form of an aluminum coating with a thickness
of - 10 pm. In order to avoid an additional dependence related to the rigidity caused by the metal, a 20-30-pm-
thick layer of the (C5H802)n plastic was applied on top of the metallized coating. The dimensions of the detector
did not exceed 50 x 10 mm.
The probing was made at 400 MHz so that at a ^-120-nsec-long emission pulse, the fine structure of the
pulse could be distinguished, that the hf noise generated during accelerator operation could be eliminated,
and that long (^-30 m) cables could be employed at relatively low losses of hf power in the measuring system.
The attenuation of the wave in the waveguide containing the ionizing gas was directly measured:
T1 (t) = In E0IE (t),
where E(t) and E0 denote the amplitude of the probing field at the output of the waveguide with and without
plasma, respectively.
The attenuation 71(t) is related to the conductivity a of the ionized air (conductivity averaged over the
volume) by the formula
TI (t) = 2, v (t),
Translated from Atomnaya Energiya, Vol. 48, No. 6, pp. 381-383, June, 1980. Original article submitted
August,20, 1979.
0038-531X/80/4806-0386$07.50 ?1980 Plenum Publishing Corporation
Declassified and Approved For Release 2013/02/01: CIA-RDP10-02196R000800030005-9
 Declassified and Approved For Release 2013/02/01: CIA-RDP10-02196R000800030005-9
Fig. 1. Normalized oscillograms of the time
dependence of the dose rate recorded with: 1)
scintillation and 2) hf detectors.
TABLE 1. Comparison of the Results of Dose
Measurements Made with the Ionization Technique
(IT) and Thermoluminescence Technique (TT)
No. oo
Dose (P) per pulse
-
IT/TT
Ratio of
expt. I
I
IT TT
peak ampls.
1
15,7
15,2
0,97
10,4
2
22,4
22,5
1,14
11,7
3
9,3
10,7
1,15
12,3
4
10,7
12,6
1,17
9,5
5
13,0
13,8
1,06
10,5
6
11,2
13,4
1,20
8,9
7
15,9
17,5
1,10
12,6
8
19,6
20,2
1,03
13,1
9
12,8
15,3
1,20
12,6
10
13,3
12,6
0,95
12,1
where L denotes the length of the plasma-filled waveguide.
It was shown in [12] that the conductivity of the air (conductivity defined by the usual constants such as
the lifetime and the mobility of the electrons) depends upon the ionization I which, in turn, according to the
Bragg-Grey theory, is related to the dose absorbed by the walls of the radiator:
D k8Srel 1.
The notation is interpreted as follows: k, coefficient depending upon the system of units employed; e, average
energy of forming an ion in the gas filling the cavity; Srel, linear stopping power of the radiator relative to
the gas.
In the case of the air-equivalent (C5H8O2)n plastic, Srel 0.92 and the dependence upon the rigidity did
not exceed 15% at E Pt~ 0.05 MeV [13]. In the source of pulsed photon emission used in our experiments, the
relative quantum yield was less than 10% at E 5 100 keV [14] which implies an additional error of