THE SOVIET JOURNAL OF ATOMIC ENERGY NO. 2

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• Document Page Count: 138
• Document Creation Date: January 4, 2017
• Document Release Date: April 3, 2013
• Sequence Number: 2
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TaMxax
:)xeprHA
Number 2, 1956
The Soviet Journal of
ATOMIC
ENERGY
IN ENGLISH TRANSLATION
CONSULTANTS BUREAU, INC.
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ATOMNAYA ENERGIYA
Academy of Sciences of the USSR
Number 2, 1958
EDITORIAL BOARD
A. I. Alikhanov, A. A. Bochvar, V. S. Fursov, V. F. Kalinin,
G. V. Kurdyumov, A. Y. Lebedinsky, I. 1, Novikov (Editor InChief),
V. V. Semenov (Executive Secretary), V. I. Veksler, A. P. Vinogradov,
N . A . V 1 a s o v (Acting Editor in Chief)
The Soviet Journal
of
ATOMIC ENERGY
Copyright, 1956
CONSULTANTS BUREAU, INC.
227 West 17th Street
New York 11, N. Y.
Printed in the United States Annual Subscription $ 75.00
Single Issue 20.00
Note: The sale of photostatic copies of any portion of this copyright translation is expressly prohibited by the
copyright owners. A complete copy of any article in the issue may be purchased from the publisher for $12.50.
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THE STUDY OF THE PHYSICAL CHARACTERISTICS OF THE REACTOR
A. K. Krasin, B. G. Dubovsky, E. Ya. Doil.n,itsyn, L. A. Matalin,
E. 1. Inyutin, A. V. Kamaev and M. N. Lantsov
In this article a description is given of experiments on the study of physical, characteristics of the reactor
of the atomic electric power station of the Academy of Sciences of the USSR. The data obtained may be util-
ized _both in connection with the starting up and operation of other similar reactors, and also in the further
improvement of the methods of engineering design calculations for the heterogeneous energy: producing water-
cooled reactors operating on thermal neutrons.
The calculation of the physical parameters of the reactor of the atomic power station [1- 3] was carried,
,out by methods which had been earlier checked experimentally only on uranium- graphite lattices with a low
content of water or steel. Therefore before the reactor was'put into operation it was necessary to check the
calculations on a reactor the construction of whose lattice cells would be close to that of the cells of the
reactor of the atomic electric power station. For this purpose there was assembled and put into operation a
graphite uranium physical reactor(GWP) which was constructed of square graphite bars of density 1.67 gm/cros.
This reactor was in the form of a cylinder of 260 cm diameter and 190 cm in height. The lower reflector was
40 cm thick, while the upper reflector was 54 cm thick.
In the active zone and in the upper reflector there were 85 vertical openings of 44 mm diameter which
formed the square reactor lattice with 140 mm pitch. In the side reflector there were horizontal openings
into which neutron 'counters and ionization chambers were placed. Mixed oxide of uranium was used as the
fissionable material (Uranium with a 10% content of 235 was used). The mixed oxide powder was placed
in the circular gap between two stainless steel tubes (mark IX18N9T) of diameter 9 x O.4 mm and 13.4 x 0.2
mm. The tubes were filled 'with the uranium oxide powder to a height of 960 mm. A heat generating ele-
ment constructed in this manner contained on the average 214 g of powder. The amount of steel in a single
element was ^, 152 grams. The inner tube was filled with water. Seven such uranium elements were assem-
bled into a single unit which simulated the working channel of the reactor. The units were loaded into the
vertical openings of the graphite pile. When the minimal critical loading with uranium was being deter-
mined,, the remaining openings were filled with graphite plugs. Before the uranium units were loaded into
the assembly, a neutron source of intensity 2. 106 neutrons/sec was placed at the center of the graphite pile,
then the whole registering apparatus was checked and the rate of counting of the counters and of the cham-
bers was determined. As the uranium loading proceeded the rate of counting increased in all the registering
equipment. If one graphically plots the variation of the reciprocal. of the relative increase in number of
counts (1/N) as a function of uranium loading, then the extrapolation of such a graph (the graph of reciprocal
multiplication) until it intersects the horizontal axis will determine to a certain degree of accuracy the value
of the critical loading. By successive extrapolation of graphs (obtained on the basis of data provided by diff-
erent registering devices) one may determine in advance with sufficient accuracy the value of the critical
loading of the reactor. The necessity for knowing the exact value of the critical mass (Mcr) is determined
by `the hazard which may arise as a result of an uncontrolled rapid increase in the power of such a reactor in
the case of even a relatively small excess of reactivity.
It should be noted that starting with a loading of'-70% (estimated from early extrapolations) it is useful
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to utilize not only the graphs of reciprocal multiplication, but also the so-called difference curve of reciprocal
multiplication- the reciprocal value of the difference in the counting rate at a given loading and the counting
rate at 70% loading N N Such a method for extrapolating the critical mass leads more smoothly to
to
the actual value of the critical loading from the
1/N
direction of smaller loadings which increases the safety of
conducting the experiment. Figure I gives graphs
of reciprocal. multiplication plotted on the basis of
readings of various registering devices. For purposes
of comparison the difference curve is also plotted
In the same figure.
The loading of the reactor proceeded from the
center outwards toward the periphery. The reactor
became critical when 54 units were loaded into it.
When the openings in the reflector were filled with
graphite plugs and uranium units were inserted into
openings intended for safety rods and control rods,
the minimum critical loading of the GWP reactor
was determined, and turned out to be equal to 50
units or `6.3 kg of U235 which corresponds to an ac-
tive zone radius of 60 cm.
90. 40 50 60
Number of loaded units
Figure 1. Curves of reciprocal multiplication.
a) Counter No. 1 on the boundary between the active
zone and the reflector; b) counter No. 2 on the boun-
dary between the active zone and the reflector; c)
ionization chamber on the boundary between the
active zone and the reflector; d) counter No. 3 in
paraffin at the outer wall of the reflector; e) diff-
erence curve.
layer of water around the rod
The theoretical calculations which had been
made by M. E. Minashin, Yu. A. Sergeev, V. Ya.
Sviridenko, and G. Ya. Rumyantsev gave (within
the limits of accuracy of the calculation) values
of Mcr (5.35 - 7.40 kg of U235 or 42- 58 units)
which agreed well with the experimental data.. On
the same reactor a study was made of the effect of
an annular layer of water (4 mm thick) on the
compensating ability of a boron control rod (26
mm in diameter) of the reactor of the atomic elec-
tric. power station. It was determined that a 4 mm
decreases its compensating ability by 10%. Calculations carried out by D. F.
Zaretsky. indicated that a 150j6 decrease should be expected. The other experiments carried out on the reactor
of the atomic electric station are described below.
Determination of the Critical Loading of the Reactor of the Atomic
Electric Power Station
On the diagram of the lattice (Figure 2) is shown the placing of the start-up registering equipment used
in the course of achieving the critical loading of the reactor. Three neutron proportional counters and three
ionization chambers were placed in the vertical channels of the reactor. In view of the fact that equipment
'designed to control the power level of the reactor of the atomic electric power station under normal operating
conditions does not .,have the sensitivity required to determine the degree of approach of the reactor to its
critical. condition,special radiotechnical equipment of high sensitivity was constructed. At: the output of one of
the circuits a dynamic speaker was placed which enabled one to determine changes in the neutron flux by a
change in the frequency of clicks.
After the insertion into the center of the reactor of a neutron source of intensity 2. 106 neutrons/ sec,:the
loading of the reactor with working channels (WC) was begun. To decrease the loss of neutrons from the un-
loaded cells of the reactor graphite plugs were inserted into them. The curves of reciprocal multiplication
were plotted from the readings of all the registering devices; in addition difference curves were also plotted.
The working channels were loaded progressively from the center out toward:. the periphery. In the course
of loading; measurements were made of the effectiveness of the action of manual control rods (MC) and of the
safety rods(SR). After 60 working channels filled with water had been loaded, a self-sustaining chain reaction
was achieved.
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Figure 2. Diagram of the lattice of the reactor of the atomic electric power station.
PC) cell of a physical experimental channel; MC) cell of a manual control rod; AC) cell of an automatic
control rod; SSR) cell of a slow safety rod; FSR) cell of a fast safety rod; T) cell of a channel for continuous
control of the temperature of the moderator ; may) boundary of the critical loading of the reactor with
water in the working channels; %-0) boundary of the critical loading of the reactor without water in the
working channels,
For practical design calculations of energy-producing reactors it is necessary to know the value of critical
loading not only when water is present in. the working channels, but also in its absence; An experiment made
without water in the working channels showed that the critical condition in this case is attained when 101
working channels are loaded. In the diagram (Figure 2) the critical loadings are marked in both cases.
Calibration of the Boron Rods and the Determination of the Available
Reserve Reactivity of the Atomic Power Station
The considerable excess of the technological loading of the reactor (128 WC) compared to the critical
loading (60 WC) is due to the necessity of having a sufficient reserve of reactivity which is required to com-
pensate for the burn-up of U235 in the course of operation during - 100 days at a power level of 30,000 kw, to
compensate for the decrease of reactivity by the waste and poisoning uranium fission products and by the nega-
tive temperature effect.
When all the working channels were loaded the excess reactivity was compensated by the complete
insertion of six boron rods of the .inner circle, four symmetrically situated rods of the outer circle, and one
automatic control rod. The eight manual control rods (MC) of the outer circle which remained not inserted,
the two safety rods(SR) and the three automatic control rods(AC) formed the reserve for stopping the chain
reaction in case of an unforeseen building up of the activity of the reactor.
The available reserve reactivity is one of the most important characteristics describing the state of
operation of the reactor, and it is therefore necessary to control its magnitude continually. To determine
the available reserve of reactivity (in relative units) a comparison was made of the effctiveness of the
influence of various portions of the control rods and of the safety rods on the reactivity of the reactor. As a
calibration unit of reactivity that reactivity was chosen which arises as a result of displacing a pair of auto-
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matte control rods by 1. cm in the middle portion of the height of the active zone of the reactor. Because the
influence of the rod on the reactivity is in this region directly proportional to its displacement, this unit is called
a lineal centimeter. To convert lineal centimeters into absolute reactivity the time of doubling of the power
level of the reactor was measured when a super-critical condition corresponding to 10, 20 and 30 lineal centi-
meters was produced. The time of doubling of the power level of a reactor determines the excess reactivity
(AK); thus it was determined that In the initial stages of operation of the reactor of the atomic electric power
station 10 lineal centimeters corresponded to OK =(4.5? 0.2) ' 1.0-1.
The influence on the reactivity of the displacement of the rod under investigation depends significantly
on the position. of the other inserted rods, and therefore it is more useful to carry out, not an individual calib-
ration of each rod, but rather the calibration of a group of similar rods after first having checked that they are
identical by comparing them under similar conditions.
Within ? 10T accuracy the rods turned out to be identical in their effectiveness. After the rods were
checked and found to be identical, they were calibrated. The rod being calibrated was set as zero (fully re-
(lineal)
moved from the reactor). The reactor was com-
pensated at a low power level in such a way that
150E ~.~ the automatic control rods would be inserted to 80
cm (to about the middle of the rod). The neigh-
boring manual control rods(MC)of the outer ring
.100 2 were removed completely. The rod being studied
was inserted to such a depth as to cause the auto-
matic withdrawal of the AC rods through 10 lineal
cm r from 80 cm to 70 cm). By means of withdraw-
d
er in-
ing a rod rar removes both from the rou un
vestigation and from the AC rods, the automatic
rem (displacement) control rods were returned to their initial position
50 100 150 200 (80 cm). Then the next segment of the rod being
a studied was inserted in such a way as to again pro-
50 /oo rsocm (lineal) duce a withdrawal of the AC rods through 10 cm.
050 too b /50 200cm (displacement) etc., until the rod under investigation had been
inserted completely. As a result of this the depen-
Figure 3. Calibration curves for automatic and man- dente of the effectiveness of the rod being studied
ual control rods(a) and a nomogram for a manual
on its depth of insertion was established.
control rod (b).
1) Automatic control rod; 2) manual control rod. Figure 3 shows calibration curves for automatic
and manual control rods and also an example is given
of the method of construction of a calibration nomogram. The linearity of the graph in the region of medium
depth of insertion of the rods confirms the appropriateness of our choice of the relative unit for measurement of
reactivity. The depth of insertion of the rod into the reactor is expressed in terms'of displacement cm as distinct
from lineal cm.
In the region of the lineal portion of the graphs the rod has the greatest effect on the reactivity. The fact
that the linear portions of the graphs for the AC and the MC rods do not coincide is explained by the fact that
at zero position the lower end of the MC rod is 60 cm above the active zone, while the lower end of the AC
rod in its zero position is level with the upper boundary of the active zone.
The effectiveness of an absorbing rod depends significantly on the position of the neighboring rods.
For example, the effectiveness of the actionof an MC rod of the outer ring is decreased. from 150 lineal cm. in
the case of removal of neighboring rods to 1.20 and 90 lineal cm . in the case of insertion of one or two neigh-
boring rods respectively. The results of the determination of the effectiveness of the action of absorbing rods
are given in Table 1.
From Table 1 it follows that the total available reserve reactivity in the initial stages of operation is
LK = 0.11 + 0.005. The error in the determination of the value of AK is made up of errors in the determination
of the identical nature of the rods, of error in the determination of the "value" of 10 lineal cm. (AK = 4.5. 10-4)
and of error introduced by the possible small fluctuations in the reactivity of the reactor during measurements.
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TABLE 1
Type of rod
One MC rod of
the inner circle
One MC rod of
the outer circle
Effectiveness of
rods in tK .. 0.013 + 0.001
For a supercritical state of the reactor not only is the effectiveness of SR rods important, but also the
speed with which they can be introduced into the active zone. This. is particularly important in the case of
a rapid increase in reactivity caused by the rupture of the thin-walled tubes which are under high water
pressure. To investigate the.operating time of the safety rods a study was made of the nature of the falling
off of the .power level with time as a result of an
emergency shut-down of the reactor. The study
it was conducted by means of recording oscillograms
r,-aM sec with the aid of an ionization chamber and an
s0r r, G,0Y0 sep oscillograph. The ionization chamber with boron
r.-t36 sec coated internal electrodes was placed into the neutron
0 0.4 A 1.0 /,G IBT,0 beam emerging from the reactor and was well, shielded
against y-rays by a layer of lead 250 mm thick. The
Figure 4. Oscillogram showing the failing off of . oscillogram showing the .falling off of the power level
the power level. (Figure 4) after prolonged operation of the reactor at a
A) The instant at which the button releasing the power level of 22,50.0 kw shows that during approxi-
safety rods is pressed. mately 1 second the power level of the reactor falls
to approximately 50%. In the case of the reactor oper-
ating at a low power level (0.01?fo) this time decreases
to 0.5 seconds. The apparent decrease in the effectiveness of the action of the safety rods in the former case.
is related to the increase of the reactor reactivity at the time of release of rods due to the cooling down of
the uranium and the water. .
The Influence of Water on the Reactor Reactivity
As has been pointed out already, the critical loading of the reactor in the case that the working channels
are filled with water is considerably smaller than in the case of the absence of water (60 and 101 working
channels respectively). This demonstrates the considerable positive influence of the water on the reactor
reactivity which had been predicted by means of calculations [3]. As a result of this it became necessary
to study the possible consequences of water entering the graphite moderator. The investigation of the influence
of water on the reactivity was carried out by means of measuring the time required for the doubling of the
power level of the reactor, and consequently of OK as the amount of water in the reactor was varied. If a
column of water in a tube (62 mm in diameter) is introduced into the central cell of the reactor its reactivity
is decreased by an amount AK = -(18 + 2) 10-4. The introduction of thin cylindrical layers of water ( 4 mm
thick) into four safety .rod channels. slightly increased: the reactivity by i ,K =+0.5 10-4.
To obtain data required for an estimate of the increase of reactivity in the case of complete flooding
of the moderator an experimental simulating channel (SC) was used with an increased water content. One
cm, of height of a standard working channel contains 3.6 cros.. of water, in the case of a complete flooding of
the reactor 1 cm of height of a cell could contain 6.6 cm3 of water, while 1 cm of height of a simulating
channel can hold 8.6 cm3 of water, which enabled one to estimate the increase in the reactivity of the reactor
in the most unfavorable case when all of the "accidental" water is concentrated close to the uranium.
A comparison of the effectiveness of a working channel (with the normal amount of water) and a simula-
ting channel with an "accidental" amount of water was made in. the various cells of the reactor with 101
working channels being loaded and for the two cases: without and with water.
In the first case the addition of water had a greater effect on the reactivity than in the second case.
As the distance from the center increased, the. effect . of. the water decreasedto.zero and even became negative.
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,lKxIO 4.
15 ' 30 45 60
Distance from center in cm.
Figure 5. Curves showing effect on reactivity.
a) One SC (100 WC without water); b) one WC with
water (100 WC without water); c) one WC. with
water (100 WC with water); d) one SC (100 WC
with water); e) recalculation of the effect of SC
in one cell (the case of accidental flooding of
the reactor).
The accuracy of the compensation of reactivity in these
experiments was OK = 3.1.0-5 .. Figure 5 shows the
results of these measurements. In order to obtain the
increment in the reactivity obtained as a result of re-
placing a working channel without water by a simula-
ting channel the ordinates of the corresponding graphs
should be added.
It is interesting to note the positive essentially
heterogeneous effect of water situated close to the
uranium. Comparison of curves a and d and also of
b and c permits one to draw the conclusion that in the
case of flooding of the moderator of the reactor there
will be a decrease in the effect of the excess water in
the cells of the reactor in comparison with that des-
cribed by case d. In the first approximation one may
assume that the ordinates of the curve d will be dimin-
ished in the same ratio as that of the ordinates of curves
c and b. The curve e of this figure obtained as a result
of such a recalculation characterizes the possible, effect
of the water on the reactivity of the reactor in the case
of the moderator being flooded.
Boundary of'
the active zone
A
Figure 6. Thermal neutron density distribution along the reactor radius.
a) Measured when six inner and three outer manual control rods are fully
inserted; b) measured when one inner and twelve outer manual control
rods are fully inserted.
Using this curve and knowing the number of working channels at each radius one may estimate the probable
total increase in reactivity under accidental flooding which turned out to be AK = 0.016, Calculationon the
basis of curve d of the excess reactivity under accidental. flooding of the moderator gives a maximal estimate
which is equal to AKmax = 0.030. It may seem from Table 1 that the safety control rods and the reserve con-
trol rods can more than compensate this increase in reactivity.
Measurement of the Probability of Resonance Neutron Capture.
As is well known, the curve for the cross-section for absorption of neutrons by Um. has sharp resonance
peaks in the epithermal region of neutron energies. An exact knowledge of the probability of resonance capture
of neutrons ( 1- gyp) is needed for the proper design of a reactor.
The measurements were carried out using the methods described in reference [4].
As a result of ten series of measurements the average value
~P = 0.906.1 0.015
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The. indicated error in the determination'of y, was principally determined by our inaccurate knowledge
of the constants ordinarily used for reactor design calculation.
A Study of the Parameters of the Neutron Distribution
The thermal neutron density distribution was studied by means of the activation of thin gold and copper
foils placed into', horizontal and . vertical experimental channels in the reactor with a subsequent measure-
ment of their 8 activity. The activation was carried out with a cadmium filter and without it. The results
of these measurements are shown In Figure 6. From the curves it may be seen that the neutron density Is
lowered at the absorber positions. As a result of analysis of such curves, the optimal depth of insertion of the
absorbing rods was chosen for various regimes of reactor operation.
For the measurement of the radial distribution of neutrons of intermediate energies the following indica-
tors were used: Indium, gold, cobalt, and manganese which show resonance neutron capture at energies of 1.45,
4.91;. 130 and 346 ev respectively. The weight of these indicators did not exceed 3 mg so that the introduction
of resonance neutron absorbers did not lead to a distortion of the resonance neutron distribution. Figure 7 shows
the results of measurements with gold and.manganese indicators. Analogous results of measurements with in-
dium and cobalt indicators are not shown here. As should be expected the decrease in the neutron density in the
region of operation of MC rods is more noticeable in the case of measurements made with gold and indium
indicators.
Since the U235 concentration is very high in the reactor of the atomic electric power station and the quan-
tities of water and of iron are also high it was necessary to study the spectrum of thermal neutrons. The spec-
trum was studied by means of a mechanical neutron selector in a vertical neutron beam emerging from the
center of the reactor operating at a power level of 1500 kw at which the moderator temperature was equal to
420 ?K.' The velocity distribution of thermal neutrons obtained do this manner is shown in Figure 8. For
comparison (dotted curve) the Maxwellian distribution of neutrons (the one most nearly corresponding to the
given spectrum) is given corresponding to a temperature of 50.0?K. From Figure 8 it may be seen that .
40
49
48
0,7
as
as
44
49
0,2
41
0
p eq
Figure 7. Radial density distribution of resonance
neutrons in the reactor.
to ie ^~
I
r ~~
I 'A
I
~.vi0'mis??
I 2 i 3 4 5 5 7
Figure 8. Spectrum of thermal
neutrons in the beam.
the temperature of the neutron gas is slightly above
500?K. The small difference between the experi-
mental and the calculated curves is related to the preferential absorption of neutrons by uranium and by. iron
in the soft portion of the spectrum. It is possible that there is a small increase in the effective temperature
of the neutron gas, in the beam connected with the escape of neutrons from the graphite surface' in the experi-
mental channel. The deviation of the.observed neutron distribution from the.Maxwellian one amounts to
not more than 10Q/6 with the measurements being accurate to not better than ? 5%.
For comparison,the effective temperature of the- neutron gas was determined by the method of boron
filters previously calibrated on the mechanical selector. Satisfactory agreement was obtained with the selector
measurements. Similar measurements of the spectrum and the temperature of the neutron gas were also carried
out for the edge of the active zone. In this case the neutron beam was taken out along the horizontal experi-
mental channel. . It was established that the temperature .of the neutron gas at the edge of the active zone ex-
ceeds the temperature of the. moderator by 70-100?C, with the difference becoming less within the indicated
temperature range. as the moderator temperature increases.
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V23S
In order to determine the magnitude of the rela-
tive contribution of resonance neutrons to the energy
released in the reactor. the cadmium ratios (RCd) were
measured for gold and for U2'5:
activation without cadmium
RCd activation with cadmium
50
/00
Rem
110
Figure 9. Cadmium ratios for U235 and A.u.
The thickness of cadmium.in these experiments was
0.25 mm. The cadmium ratio for U2ss determines the
relative number of fissions in the thermal and resonance
regions. From the cadmium ratio for gold, the ratio of
fluxes of thermal and resonance neutrons in the reactor
was determined. At.the center of the reactor the cad-
mium ratios for Au and for U235 are respectively equal
to 1.5 and 12. From the curves giving the dependence
of the cadmium ratios on the reactor radius (Figure 9)
it may be seen that 8.3% of the U2s5 fissions take place in the epicadmium region.
1. Critical loading in the presence of water
in working channels,- expressed as a.num-
ber of WC.
2. Critical loading in the absence of water
.in working channels, expressed as a num-
ber of WC.
3. Reserve of excess reactivity at the begin-
ning of operation.
4. Compensating ability .of:
one MC rod of the inner ring
one MC rod of the outer ring
two SR rods
5. Probability of escaping resonance capture
6. Fraction of U235 fissions in the epicadmium
region
7. Temperature of the neutron gas.
Experimental value
99
0.013 ? 0.001
0.12
0.007_t 0.001
:0.007
0.018 ? 0.002,
0.02
8.3%
At the nominal power level
exceeds the temperature of
the medium by 70?.C.
CONCLUSION
The experimental data obtained above permitted us to check the results of design calculations for the
reactor of the atomic electric power station. The heterogeneous effect has been demonstrated of the influence
of water on the reactor reactivity which increases noticeably as the amount of water close to the uranium is
increased. Water which is situated far from the reactor decreases reactor reactivity. In the most unfavorable
case of the complete flooding of the reactor moderator with water, excess reactivity appears for the compen-
sation of which two safety rods and the reserve control rods are sufficient. Experimental investigation of the
neutron density distribution along the radius and along the height of the reactor for various rod positions per-
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mitted us to select the optimal system for the compensation of excess reactivity by control rods. In Table 2
are given the principal results of the investigation of the physical characteristics of the reactor of the atomic
electric power station. For comparison calculated values [3] are also given.
The following persons participated in the present work in addition to the authors: M. V. Bakhtina,
Yu. Yu. Glazkov, V. Ya. Kitaev, Yu. I. Koryakin, E. F. Makarov, V. A. Parfenov, L. P. Khatyanov,,, V. R.
Trubnikov and Yu. Yu. Shuvalov.
The authors consider it their duty to express their deep gratitude to Professor D. I. Blokhintsev for the
general scientific direction and for the valuable advice given in the course of carrying out the present work,
and also to academicians I. V. Kurchatov, A. P. Aleksandrov, A. I. Alikhanov, corresponding member of the
Academy of Sciences of the USSR N. A. Dollezhal' and to Professor V. S. Fursov for taking part in the discussion
of the planning of experiments at the time of starting up the reactor. The authors also wish to thank the personnel
of the atomic electric power station for aid in carrying out experiments, and in particular their thanks go to
N. A. Nikolaev, A. N. Grigorjrants? G. N. Ushakov and V. A. Konovalov.
LITERATURE CITED
[1] D. I. Blokhintsev, N.~,A. Nikolaev, Reactor Construction andReactorTheory. (Reports of the Soviet
delegation at the International Conference on the.Peaceful Uses of.Atotnic Energy). USSR Acad. Sci; Press
1955; p. 3.
[2] D. I. Blokhintsev, N. A. Dollezhal', A. K. Krasin, Atomic Energy ,1956,, No. 1, 1Q4T.p. 7)0.
.[3] D. I. Blokhintsev, M. E. Minashin, Yu. A. Sergeev, Atomic Energy :.1.956; No.. 1, 24 (T.p. 21)?.
[4] M. B. Egiazarov, V. S. Dikarev, V. G. Madeev. Measurement of the Resonance Absorption of Neutrons
in a Uranium-graphite Lattice. (Report at the session of the Academy of Sciences of the USSR devoted to the
peaceful utilization of atomic energy, July 1-5, 1955).
T. P. = Consultants Bureau Translation pagination.
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MULTIGROUP METHOD OF CALCULATIONS USED IN THE DESIGN OF THE REACTOR
FOR THE ATOMIC ELECTRIC POWER STATION
In reference [1] a survey is given of the methods and of the results of the
physical design. calculations for the reactor of the APS; which are closely related
to the well known two-group method. The design calculations for the APS reac-
tor were also performed by means cf the multigroup method. Some of the results
-of these calculations obtained on the basis of an extensive use of the method of
finite differences are given in the present article.
1. Statement of the Problem
The principal problems of the physical design of a reactor are the determination of its critical size and
the determination of the spatial and energy distribution of the neutron flux and of the neutron weighting func-
tion.
A more or less satisfactory solution of these problems is possible within the framework of the well-devel-
oped age-velocity theory which, as is well known, agrees sufficiently well with experiment only in that case
when the elastic slowing down of the neutrons takes place on nuclei whose mass is considerably larger than the
mass of the neutron.
If ,in the moderator mixture, nuclei of hydrogen-containing materials, are present then the use of age-
velocity theory may lead to significant errors in the determination of the spectrum and of the neutron: weight-
ing-function and consequently of the critical size of the reactor.
In uranium- graphite- water reactors, water is usually the principal heat transfer medium and it is placed
near the heat producing elements forming together with them and with the surrounding graphite moderator
the basic cell of the heterogeneous reactor.
The reactor of the atomic electric power station belongs exactly to this type of thermal neutron hetero-
geneous reactors [2].
It is evident that the closer the water in the cell Is situated to the .uranium lumps, the greater number of
neutrons will be slowed down in the water and consequently the water will be more effective in its role as mo-
derator. This essentially determines the basic heterogeneous effect of the water on the slowing down of neu-
trons in the cell of the reactor.
As regards the region of diffusion of thermal neutrons, here it is very important to note the fact that at
thermal energies water has a.very'considerable capture cross section. However, in spite. of the existence of
considerable absorption of neutrons by water in the region of thermal. neutron diffusion estimates, ce.nfirmed
by experiment, show that. its total effect on the reactivity of the reactor of the atomic electric power station
is positive. At the same time one should also note that the effect of water situated far from the uranium
lumps may turn out to be negative because of the strong capture by water of neutrons in the region of thermal
neutron diffusion.
A proper method of taking into account all the competing factors allows one to choose in the most advan-
tageous manner the construction of the cell of a heterogeneous reactor.
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Consequently, one of the main problems that must be solved prior to the calculation of critical size, is
the design of the cell of the heterogeneous reactor.. These calculations allow one at a later stage to determine
physical constants averaged over the neutron spectrum in the cell, the effective age of neutrons as a function
of their energy, and the probability of escaping resonance capture in the course of the slowing down process.
One should note, however, that particular difficulties arise in the solution of Just.this last problem.
After all the effects indicated above have been estimated on the basis of considering a single cell and
after all the effective constants averaged over the spectrum of the cell have been obtained,the heterogeneous
reactor may be replaced by a homogeneous one ? calculations on which represent a much simpler problem,
and may be carried out by several different methods.
2. The Averaging of the Physical Constants for the Reactor
of the Atomic Electric Power Station
In this section we must first of all solve the problem of the spatial and energetic distribution of neutrons
Figure 1. The reactor cell used for the calculation.
a) The reactor lattice; b) the equivalent cell.
(over the cell of a heterogeneous reactor)..
In view of the fact that the geometry of the
cell of the active zone of the reactor turned out to be
quite complicated it was decided for purposes of cal-
culation to replace the actual geometry by a certain
effective one possessing axial symmetry (Figure 1).
Thus, it was assumed that zone 1 of the cell is filled
with water, zone 2 contains a mixture of uranium,
water, graphite and structural materials, and finally that
zone 3 is filled with graphite.
In order to calculate the neutron spectrum in the
cell the well known kinetic equation was used in the
Pi- approximation.
T_ r ar, r4't + 4)o (r, u) du'fot (u.- u') 4)0 (r, u') -{- S (r, u),
i u-qt
u
3~ 9 as 00 -f ~t (r, u)= (2,1)
du'f rt (u - u') 4)r (r: u'),
{ uqt
where 00 and 01 are the coefficients of the first two terms in the expansion of the collision density function
-ri(r, u, g) = nvZ5 in terms of Legendre polynomials:
4) (r, u, R) _ 4o (r, u) + ~i (r, u) Pl (?).+ .. .
(Mi I f)a u M'+1 -Z-M{-1 ~ 1 (A)
4M1e P~f2 e 2 e J,
qi is the maximum logarithmic energy loss for a neutron colliding elastically with a nucleus of mass Mi;
S(r, u)..are the monochromatic sources of neutrons of energy 2 Mev ( u = 0) which uniformly fill zone 2 of
the cell.
For all the elements i with the exception of hydrogen the well known hypotheses and methods of the
age-velocity theory were used for the evaluation of the integrals in the right-hand sides of equations (2.1).
The system of the integers-differential equations so obtained was then reduced to a system of multiple-
group diffusion equations. In doing this it was assumed that within each group the collision density 0 (r, u, p)
? With regard to a more accurate method.of taking, into account the heterogeneous nature of the installation
see reference (3).
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may at a fixed value of r be represented as a linear function of u.
4) (r, u, p) ? ~t (r, P) 4-trl+1(r, l~)(u-U.d;
where s1(r, ?) ,(r, ut, p) and ui are certain representative points over the whole range of lethargy selected
for the purposes of calculation.
In order to be able to take into account certain conditions at the zone boundaries inside the cell, the
system of diffusion equations referred to above was replaced by a finite difference system which iris not difficult
The results of calculatidns of the neutron spectrum nv(r) over the reactor cell are given in Figure 2.
After the neutron spectrum in the reactor cell has been found,the averaging of the physical constants
does'not present any difficulties and may be carried out for any logarithmic energyU by means of the
formula
S X (r, u) nv dv
X (u)=(D)
S nu dv
(D)
where (D) is the total volume of the cell, while 9 Es, Ec, Ef and 1 = D should be used in turn for the
Etr
quantity X (r, u). After the neutron spectrum in the cell had been found, the calculation of the effective
age of the neutrons was carried out with the aid of the formulae
2
Figure 2. Spatial and energetic neutron distribution
over the reactor cell. Figure 3.Effective neutron age.
1) Q(r); 2) Nv(r); 3, 4, 5)nv(r, u) for u = 2.5; u = 1) r(u) for a homogeneous mixture; 2) r(u) taking
14.0; u = 17.5 respectively; 6) 10-14(r); into account the spatial and energetic neutron dis-
1, II, III) zone boundaries within the cell. tribution in the cell.
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Nu du u _
no
du
?
3Zp (r, 0) 1 .9`~t.
3Ez
Nu du J ww nu du
1k
(
)
where Nv is the neutron flux before slowing down. E is the mean logarithmic energy loss per collision.
On comparing results of calculations carried out with the aid of (2.3) with the value of 7(u) obtained
from the formula for a homogeneous mixture?
U
du
ti (u) + `
A 31280 M,Ej P
0
where Eso = ?,28(0) and 1 are quantities calculated taking into account the formal replacement
3
EsEtr
of the cell by a homogeneous one, it becomes evident that the effect of,heterogeneity in the slowing down
process is sufficiently small, and it may be entirely neglected.( Figure 3)..
This circumstance shows in particular that it is possible to use the age-velocity slowing down theory
for the calculation of the critical size of the reactor.
We now proceed to one of the most essential characteristics of a heterogeneous reactor- the probability
that a neutron escapes resonance capture by U238 , nuclei in the course of the slowing down process. This quan-
tity is denoted by cp.
In view of the weak mutual shielding of uranium lumps in an actual reactor cell, the calculation of
reduces to the investigation of a single annular block cooled on the inside by water, and surrounded on the out-
side by moderator 3 of infinite extent (Figure 4).
Figute 4. Annular block of a single cell.
1) Water; 2) fuel; 3) moderator.
Let us assume that the flux of neutrons with ener-
gies above resonance is essentially determined by the
slowing-down spectrum and varies but little from
point to point.
Then, in accordance with the work of Gurevich
and Pomeranchuk [4], it is not difficult to determine
the total number of those neutrons being absorbed
which come from moderators 1 and 3 towards the
uranium block, and in order to do this it is necessary
to calculate the attenuation of their flux along
various directions of their motion.
The calculation of the attenuation of the neutron flux along all possible directions and the subsequent
integration over all these directions gives us the possibility of obtaining the total number of neutrons of a
given energy being absorbed in the uranium block.
The subsequent summation of the number of absorbed neutrons over all the energies taking into account
the shape of the resonance leads to a formula which holds when the thickness of the uranium blocks is much
smaller than the mean free path for scattering in the material of the block."
In rp .. no A JSx t/11 + Ss V 1, 1 1~+ BV,~s (2.5)
tV8?E8 ,
Formula (2.6) for the cell of the reactor of the atomic electric power station was obtained by V. V. Orlov.
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where n0 is the number of uranium blocs per cell; Vu is the volume of a uranium block;, a is the porosity of
the uranium; Si and S2 are the inner and outer surfaces of the block; Vs is the volume of the moderator in
the cell; E is the mean logarithmic energy loss; -1-1, _ TDF1(d ' f`2' N ~I where d and D are
respectively the inner and outer diameters' of the block and Fl( /D) and F2( D) are certain functions whose
graphs are shown in Figures 5a and 5b.
0 0.2 04 0.6 08 I,0
,Ta
Figure 5a. Graph of the function F1(a, 0) where
d
a = d (E sH~ for a resonance energy 8 = D
A and B are constants evaluated from formulae:
_. t1 ?oi f
L.I Ef
0(
dE
)E,
where oo). is the absorption cross section at the peak of the j'th resonance, and E. As the energy of this reson-
ance,. the summation over j being extended to all the resonances above the energy E.
To determine the values of the constants A. and B over the whole spectrum theoretically it is necessary
to have an accurate knowledge of all the parameters for all the resonances of U. In view of the inadequate
knowledge of these parameters, V. V. Orlov used the experimental data on the total absorption of neutrons
by nuclei of U238 obtained for solid uranium blocks, of various diameters [5].
In reference [5], constants A and B for the formula of Gurevich and Pomeranchuk [4]
AS IfT If-. + BVue
In?= - . EEevg ,
(2.6)
where _ D ? F,(0). are found for solid uranium blocks of diameter D placed inside a moderator.
It is evident that as d -*- 0, Formula (2.5) goes over into (2.6). Consequently, the values of the constants
A and B in (2.5) and (2.6) must coincide. Thus Formula (2.5) allows us to determine the magnitude 'of the
probability of escaping resonance capture by U2ss nuclei in the reactor cell. For practical calculation it is
useful to effectively distribute in energy the quantity so obtained among the strongest absorption resonances
of U238.
0,5
f -0.8
04 0.6 68_ 1..0
Figure 5b.. Graph of the function F2(a, Q)? -
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By a similar method, but taking geometry into account, one may calculate the quantity p - the fast
neutron multiplication factor. Since for the reactor of the atomic electric power station p is close to 1, we
shall not take it into account in the. following for the calculation of critical masses.
3. Basic Equations
We consider the system of basic equations for the homogeneous reactor equivalent to the reactor of the
atomic electric power station:
VDO?VNv-Ee0?Nv
UT
-of { J nvE1du+fiE/T}
0
E,ONv=q(r, 0), (3.1)
VD ?Vnv-EC?nv(r,. u)=au
VDT ? VT EcT4 (r) = - q (r, UT), 1
where q = nvEc is the slowing down density; 4)(r) is the thermal neutron flux; v f = 2.43.
Quantities relating to the thermal energy region are denoted by.the index "t".
As is well known, Equations (3.1) hold in the case when :the absorption cross section does not depend
strongly on the energy.
If Ec has a resonance at energy u, then the slowing down density q(r, u) changes discontinuously on
passing from energy u +.0 to u- 0 by the amount
q(r, u+0)=rp(u)q(r, u-0), (E)
where rp is the probability of escaping resonance capture at the given point.
Together with the boundary condition, which in this case must be taken to require the vanishing of the
solution at the extrapolated boundary, the eigenvalue problem becomes closed.
The solution of the system (3.1) is found as a rule by the method of successive approximations which
consists of assuming a certain source distribution Q(.t') and of finding its new value in a better approximation
by means of solving the system of Equations (3.1).
This process should be continued.until the ratios of two consecutive values of the function.Q(i) become
equal within the prescribed limits of accuracy to the same constant for all points of the active zone. The
ratio found by this procedure will be the desired eigenvalue of the system, i.e.,
keff. (n- 11
where the index n is the order of the successive approximation.
Thus the solution of the eigenvalue problem has been reduced to the successive solution of Cauchy
problems which are much simpler from the point. of view of actual specific calculations.
In addition to the determination of the neutron spectrum, the calculation of the spatial and energy
distribution of the neutron weighting function in the reactor is also of considerable interest,
In this case, as is well known, the mathematical problem reduces to the solution of the following
system of differential equations
Q (r)
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'VVDA0 ? V Nv* -? ERn ? Nv'" (r) _ - +u ?.rtv* (r, 0),
7.1) nv 11V* (r, u.) Nv* (r),
nv* (r, u,T) = ~~'T (r),
VDT . V 1)* - E~T,Lr* (r) _ v/~ fTN0* (r)'.
there, just as in the case of the basic reactor Equations (3.1),it Is necessary to assume that the solution will be
bounded inside the reactor and equal to zero at the extrapolated boundary.
The method of successive approximations for the solution of the system of Equations (3.3) consists of the
following. In the right hand sides of Equations (3.3) the function Nv* (r) is given, and then a new value of
NO (r) is found as a result of solving the system (3.3) etc. In this procedure keff is defined as the ratio of
two successive values of Nu * (r) which have become established after several interactions:
k _ Nv*(n)(r (3.4)
eff. ^ Nv? (n- 1)(i)
It is naturally understodd that the eigenvalues of the basic Equations (3.1) must coincide with those of the con-
jugate Equations (3.3). It is useful to keep this circumstance in mind when estimating the accuracy of the
calculations performed.
Figure 6. Diagram of the reactor.
Figure 7. Reactor types used for calculations.
The system of E quations (3.1) as well as the system (3.2) may be solved by means of various approximate
methods. But before proceeding to the solution of these systems one should note one. important circumstance.
Because the reactor of the atomic electric power station has a complicated geometry (Figure 6), the
direct solution of systems (3.1) and (3.3) is very difficult. Therefore, in order to simplify the solution of
these systems of equations usually the reactor of cylindrical shape is replaced by a certain spherical one of
an appropriate effective size. However, in our case this method may lead to considerable error.
It turned out that the most flexible method for the solution of problems was the method of successive
approximations, which consists of the following.
Let us consider a cylindrical reactor without end reflectors which has the same radius as the real one
(Figure 7). Assuming a certain effective height Heq of this reactor and determining keff., we can then find
the extrapolated boundary of the equivalent reactor without side reflectors.
R =R+oR
eq
where 8R is the equivalent increment.
Let us then consider another one - dimensional reactor whose height corresponds to the actual one and
whose side reflectors are replaced by the equivalent increment found above (Figure 7). Having determined
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keff, for this reactor we can then calculate the equivalent increment of height, etc.
These calculations should be continued until a limiting value of keff has been established which will be
the required approximate eigenvalue of the system. In order to find the critical mass of the reactor, one must
carry out a series of analogous calculations either for different dimensions of the active zone and a fixed concen.
tration of UPS, or for different concentrations of U236 and fixed reactor dimensions. The desired value of the
critical mass will be obtained for that uranium concentration which gives keff. - 1.
4. The Multigroup Method of Solving the Equations
In the preceding section it was shown how the problem of calculating the critical mass of the reactor can
be reduced to the successive solution of one-dimensional plane and cylindrical problems.
Since the system (3.1) as well as (3.3) consists of equations of the diffusion and age-velocity type, let us
investigate the approximate methods available for their solution.
For the active zone of one-dimensional reactors with plane and cylindrical geometry, the age-velocity
equation of the system (3.1) has, as is well known, the form
DVaq Ecq-89
R_. 8~'8 du
0 for plane geometry
1 for cylindrical geometry.
V2 e,1QB
Or . Or
In solving Equation (4.1) for the various zones of the reactor, the following boundary conditions must be
used at the boundaries separating the different media
fE8 q
D ~y D'
GEB Y q - ~;~, Vq
(primes denote quantities referring to adjoining zones in the reactor).
At the same time one must assume that the solution should remain bounded within the reactor volume,
and must vanish over the outer extrapolated boundary of the reactor.
,We shall seek the solution of.Equation (4.1) by means of the method of nets..* In order to do this we
shall superimpose on the semi-infinite strip (Risr 0) within which the function q(r, u) is defined as a
rectangular net with a pitch it along the r axis, and a pitch An along the u axis.
It is well known that a differential operator may be replaced by a number of. approximate methods.
However,for making calculations for thermal neutron reactors or other similar ones a triangular scheme is
the most convenient one, which may be represented in the case of weak absorption during the slowing down
process in the form
" The multigroup method of reactor calculations described in this article. was developed by the author in
1953 [6].
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i r BTa
+i
+ ?
M (
)
qL
qk
9k
,
A'C
2
Lc
2La
i Eo
ul4i D du
Bs
IF
D
q+t
UI
2qh {" qk_9
for
a =
0
M,(9)
,
qk+ _.. 2qh+qk-i Zrk(9
-9
We must now formulate the subsidiary relationships which will enable us to obtain in a similar way the solution
at the zone boundaries and at the center of the .reactor. For this, the radius of the active zone must be divided,
up into segments of length Or in such a way that the point r = 0 should fall in the center of the first interval
Or, the boundary between the active zone and the reflector r = R1 should fall in the center of the last interval,
while the reflector must be broken up into segments Ar' in such a way that the boundary between the active
zone and the reflector, and .also between it and the second adjacent reflector, must fall at the center of the first
(or the last) interval 4r'.
Thus it is assumed that the solution of the problem for each zone may be continued analytically by half
of a Ar step in the direction of the adjacent. zones. This well known method [7] turned out to be convenient,
for specifying the conditions at the zone boundaries in the finite difference formulation.
Indeed, if we assume that Xj + 1 and Yj + 1 are the desired functiohs at a fictitious point, then we may
obtain without difficulty that
X1+1-K1 1+1 . L' 1+1
qa + qa+ i'
Y1+1
=1 [X1*1.-}-qe?1]._qe
where K1 L1= ~-?- 2 9
Es Br D' 8E
(F)
VIP E'F+e
.and the index "s" is the number of the last point in the. active zone of the reactor (the primes on the function
qk have been omitted).
For the boundary condition at r = 0 it is necessary to take
qI f' - qJ+4.'
ql+1 ?qi+1 (.2r)
Assuming further that qn= 0 (where qn= qj(R) and.R is the extrapolated reactor boundary) we arrive at a
closed system of difference equations with the aid of which one may 'find the values of the functions at all the
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q1-+i = q1 +1
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fundamental and fictitious points of the reactor. Since the difference Equation (4.3) has already been solved with
respect. to the unknown function qk+ 1 the system may be solved in terms of the simplest arithmetical operations.
The successive solution of the system referred to above for all j 1, 2, ... m leads us to the desired solution
of the slowing down problem.
The diffusion equations of the system (3.3) are solved in an analogous manner.
Let us consider one of the diffusion equations
LCT
where f(r) is a given function; 1/Lst is a constant which has different values in different reactor zones.
As before, let us assume that
d(I I = 0,
dr ,r-0
Ro)=0,
fi = (1)'
d4 d4)'
DT dr - DT dr
In the finite difference form the problem ( 4.6, 4.7 ) in analogy with the earlier case takes the form
r a
M (`D1a) __ D! I)h = - Q T2/h
(k=1, 2, ...,.s
I)1, (D? = 0,
X = Kc3 -}- LII'8+1, Y = X -f - 4) - (De +1+
1-A 2) DT 6r
K=11+X, L=i+X' DT Ar,
(4.7)
For convenience in solving equations ( 4.5-4.7 ) we eliminate from systems of the form (4.8) the unknowns
X and Y with the aid of (4.10). As a result we obtain for all points of the reactor the following equations:
ah.4)h+1. - bk(bh + Ch4)h-1 = -- '6Tk /,h'
Dividing equation (4.11) by ak we shall obtain the difference system of diffusion equations in the final form
q) h+1 - Bh I h + Ch(I)h-1 Eh{/ h 0.12)
with the condition that
111_1= 4)19 'I1? = 0.
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In (4.12) the following notation has been adopted
bh
C3h=ah,
(V + Vk+1) (Ih+1 + ah+1 4 h+1) _ Ehf hl (H)
ch Aax
Ch = aft , 8~i = ah
We now go to the solution of the system ( 4.12-4.1'3 ). In this connection it is useful to examine basically
two methods which are the most. applicable from a computational point of view. The first of these amounts to
a method of reducing the boundary value problem to two Cauchy problems. For this it is sufficient to specify
two different values of the unknown function c at the junction point k = -1 under the condition 'Z'.. = 4'1?
As a result of the successive solution of the system (4.12) we shall correspondingly obtain two linearly
independent solutions. Constructing a linear combination of these two results which is a solution of Equation
(4.12) and which satisfies condition (4.13) we arrive at the desired solution of the problem.
This method turns out to be very convenient in a number of cases. But for a large number of computational
points the rounding-off error which increases exponentially from point to point leads to a significant loss of accu-
racy, and this leads to the necessity of carrying out the calculation to a larger number of significant figures.
In such cases it is useful to utilize a method proposed by I. M. Gelfand: and 0. V. Lokutsevsky (10) and
independently by Stark [8] which may be called the method of difference factorization. It consists of writing
the second order difference equation (4.12) in the form of a system of three first order difference equations.
Thus in addition to (4.12) let us also consider the following:
'h+1- ")h+11I'- Ch+11h,
V~h - +h
(4.15)
and the constants. ?k and ok are so far arbitrary; they must be so chosen that equation (4.13) becomes identically
the same as (4.12). Fairly simple manipulations lead to the system of first order difference equations:
Ch+1
Yh+1 Bit -Ph
Zh+1 Rh+1 [Zh + Eh fh]
~ > _ Ph+1 0h+1+ Zh+1
h Ch+1
The new quantities 5k and Zk are related -to IN and ok by the relations:
Zh= Ih I.C1-Ph)Oh.
The computation of Sk and Zk from Formulas (4.16) is carried. out, from left to right, and the computation of
11)k from right to left. Such computations do not lead to an exponential increase in the rounding-off error and
therefore this method is free of disadvantages inherent in other methods.
With respect to the boundary conditions at the center of the reactor and at the -outer extrapolated boundary,
it may be easily seen that they will be satisfied if one assumes
01='CV Zi = O, (DN.= N.O.
14,18)
In conclusion it should be noted that in certain cases it is useful to reduce the age-velocity equatiop to a
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system of equations of the diffusion type and to carry out the solution by the method of difference factorization
which possesses exceptional universality and applicability. This method may be easily programmed on the
majority of computing machines.
20
Figure 8 shows the results of the determination
of the neutron spectrum for the critical state of the
S
reactor of the atomic electric power station.
Figure 9 shows the results of calculating the neu-
trop weighting function in the reactor.
s
0.75
n
Figure 8. Spatial and energy distribution of neutrons
in a critical reactor.
1) 10 Q(r); 2) nv(r); 3, 4, 5) nv(ir, u); 6) 4r(t) for u =
=1.0; u = 5.0; and u = 17.75. respectively; I and II )
boundaries of reactor zones.
75 SO 75 NO
125 /SO
The value of keff. computed with the aid of Figure 9. The spatial and energy distribution of the con-
the fundamental reactor equations is equal to 1.023, jugate function in the reactor.
while the same value is computed with the aid of the 1) e (r); 2, 3) nvx(r, u) at energies u = 8.0; u = 1.0
conjugate equations is 1.021. Thus, keff. computed respectively; 4) Nv' (r).
with the aid of the fundamental and the conjugate
reactor equations differs by 0.2%.. This circumstance points to the fact that the mathematical problem has been
solved with good accuracy. The total error in the determination of the critical parameter of the system keff.
amounts to 2.3%. The calculated value of the cadmium ratio is in good agreement with the experimental value.
The data required as the point of departure for the multigroup calculations came from reference [1].
The author expresses his deep gratitude.to D. I. Blokhintsev and also to E. S. Kuznetsov for discussions of
this work and for valuable remarks. The author is also indebted to V. V. Smelov who performed all the compu-
tations and who carried out the design calculations on a reactor cell.
[1] D. I. Blokhintsev, M. E. Minashin, Yu. A. Sergeev, Atomic Energy .1956?No. 1, 24(.T.p. 21)*.
[2] D. I. Blokhintsev, N. A. Nikolaev,"Reactor construction and reactor theory'? (Reports of the Soviet
Delegation at the International Conference on the Peaceful Uses of Atomic Energy). Acad. Sci. USSR Press;.
1955? p. 3.
[3] $. M.. Feinberg.'" Heterogeneous methods of reactor calculations. Survey of results and comparison with
experiment.", Ibid. , p._ 152.
"
[4] I. I. Gurevich, I. Ya. Pomeranchuk, The '.theory of resonance absorption in heterogeneous systems.*,
[5] N. B. Egiazarov, V. S. Dikarev, V. G. Madeev, Measurements oft resonance neutron. absorption in
uranium-graphite lattices," (Report at a session of the Acad. Sci. USSR, 1955).
[6] G. I. Marchuk, On approximate methods of nuclear reactor calculations., (Presentation at a session of
the Acad. Sci. USSR, 1955). .
[7] L. Kollats," Numerical methods of solving differential equations" Foreign Lit. Press; 1953.
* T. P. = Consultants Bureau Translation pagination.
0.5
~
f
Q
3 ~
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[8] R. Erlich, H. Hurwitz, Nucleonics 12, 2, 23 (1954).
[9] S. Glasstone and M. Edlund, The Elements of Nuclear Reactor Theory, Foreigh Lit. Press, 1954.
[10] 0. V. Lokutsievsky, Report it a Conference of Functional Analysis, Moscow, 1956.
161
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MASSES OF THE H, D, He4 AND C12 ISOTOPES
R. A. Demirkhanov, T. I. Gutkin, V. V. Dorokhov, A. D. Rudenko
Results are given of the measurements of the masses of the H, D, He4 and C12 isotopes carried out by
means of a mass -spectrograph. with a resolving power of 70,000-100,000. The data obtained agree well
with the corresponding mass values obtained from the energy balance of nuclear reactions.
Introduction
A careful measurement of isotopic masses is one of the best means for determining such an important
property of the nucleus as the binding energy of the nucleons in the nucleus. The masses of the isotopes of
medium and heavy elements are in most cases not known with sufficient accuracy, or have not been measured
at all. Therefore their measurement is of considerable interest in connection with the necessity of the experi-
mental determination of the binding energy of the nucleons in the nucleus, particularly in the region of the
"magic numbers" for medium and heavy nuclei.
The mass spectrographic determination of the masses of the medium and heavy elements is carried
out by means of utilizing intermediate doublets connecting the isotope being measured with Ots
In most cases the intermediate doublets are made up of ions of the various combinations of the iso-
topes H, D, C12 (C12 H or Ct2 D ), and therefore the accuracy of measurement of the masses of the isotopes
m n m n
of medium and heavy elements is determined first of all by the experimental error in the measurement of
the masses of the H, D, C12 isotopes.
During the.last few years a number of articles have appeared devoted to the precision measurements
of the masses of the isotopes.of the light elements by means of mass-spectrographs [ 1-5 ]. Moreover, on
the basis of data on the total energies of nuclear reactions isotopic masses have been calculated with con-
siderable accuracy [6, 7].
The values of the masses of H and D obtained by various authors agree well with each other, while
values obtained for the mass of C12 have a spread far outside the limits of the quoted experimental errors
(see Table 3).
The lack of reliable data for the mass of the C12 isotope excludes the possibility of using ions of the
group C11~21 Hn for a precise calibration of the mass spectrographic scale in the region of medium and large
masses.
This fact led us to make new, measurements of the masses of the H, D, He4 and C12 isotopes.
Description of the Apparatus
A schematic diagram of the mass -spectrographic equipment which was used for the measurement of
the masses is given in Fig. 1. The apparatus was designed and constructed by M. Ardenne with the col-
laboration of G. Jaeger and the authors of the present article. The ion-optical arrangement is a variant
of the Bainbridge -Jordan system.
The ion beam was produced in the following manner: the ions are generated in a plasma source
with ordinary charge contraction. The voltage of the discharge arc was 50-200 volts with the arc current
equal to 0.25-0.75 amps. The ions are removed through a circular (0.15 mm diameter) emission opening
So in the anode. The magnitude of the accelerating voltage is 35-45 kv. The entrance slit S1 of width
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Fig. 1. Schematic drawing of the mass-spectrograph.
I) Plasma ion source; II) cylindrical electrostatic lens; III) magnetic analyzer; IV) photographic
camera with an ion-optical converter; V) diffusion pumps; So-emitting slit; Si- principal en-
trance slit; S2- aperture slit; Ss- intermediate slit; S4-entrance slit into the magnetic analyzer.
5-10? is at a distance of 10 mm from the emitting slit (So). At a distance of 212 mm from the exit slit is
situated the aperture slit S2 of dimensions 30 x 1000 ?. Thus the aperture of the beam was -' 7 x 10-6, The
mean radius of curvature of the cylindrical condenser is equal to 300 mm, the angle of bending is 4>1 = 63?38'.
In the plane of the image of the cylindrical lens there is a diaphragm with a slit Ss of dimensions 200
to 5O0?. The entrance slit S4 of the ion beam in the magnetic analyzer has the dimensions 500 x1OOO?- The
gap between the magnet. ; poles is 3 mm. The radius of curvature for the principal mass mo In the magnetic
field is equal to 300 mm. The bending angle is 4>m = 60?. The cardinal points of the electric and the magnetic
lenses are situated symmetrically. The plane of the photographic plate is at an angle of 30? to the optical
axis.
. The source of high voltage for the acceleration of the ions is a half-wave rectifier with a large time-con-
stant. The potential is applied to the plates of the cylindrical condenser from dry "B" batteries BAS-G-80-L-2.1
specially selected and periodically checked for stability. The voltage of the order of 2000-3000 volts is ap-
plied symmetrically so that the surface of zero potential passes in the middle between the condenser plates.
The electromagnetic windings are fed from storage batteries of 135 ampere-hour capacity. The operating pres-
sure in the region of the cylindrical condenser is 5-6 x 10-6 mm of mercury.
The average resolving power determined from the line half-width measured on a comparator is 70,000.
The maximum resolving power obtained for the C12 HE-NM doublet as a result both of comparator and of micro-
photometric measurements amounts to 100,000 for. the C'2HE line, and to 120,000 for the NU line. The lines
were recorded on "Schumann" photographic plates ofdimensions 6 x 18 cm. 14 spectra were photographed on
each plate.
The use of a plasma source with a high current density of emitted ions (~50 mm/cmt) and with a small
spread in the velocity of the ions (_ 2 v) allows one to make measurements with a small beam aperture, and
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Fig. 2. External View of the equipment.
with relatively short exposures (0.1-5 see). With such
short. exposures the requirements on the stability of the
equipment supplying voltage to the magnet and to the
cylindrical condenser are reduced, while the small
value of the aperture and the small velocity spread of
the ions (Au/ u - 5 x 10-5) ensure a small value for
errors of the second order and a good sharpness of the
image.
The ion optical converter which is situated very
close to the plane of the photographic plate enables one
to observe visually the mass-spectrum, to adjust the group
of lines being observed in the center of the focal plane,
and to monitor the quality of the spectrum during expos -
tire by observing the outer lines. Special provisions for
adjusting the ion-optical system (quartz plates with a
fluorescent layer, observation windows, means for reg-
ulating the width and position of the slits in the pumped-
out system), and also vacuum locks for changing the
photoplates and for separating the source from the vacuum system of the apparatus permit one to prepare the
mass spectrograph for operation in a relatively short time.
Dispersion and the Mass Scale
A mass spectrum always contains lines whose ions differ from each other in mass by the magnitude of the
mass of the hydrogen atom M(H) = 1.008142 + 2 x 10-6 a.m.u., e. g. N14+ _ N14H+ _ N14H2 _ N14H3
Having selected certain of these lines as "base-lines" one may experimentally calibrate the mass scale,
i.e., for a given region of the photographic plate one may determine AM (Ox,x) in the following approximate
form:
AM = A (ox) + B (Ax)2 +
If the spectrogram contains "base-lines" with a known mass difference M1 and M2 situated at positions x1 and
x2 (Fig. 3), then the constants A and B may be found from the following relations:
Mtx22-M2xi
A=
xrx2-x2,xi
B - M2x21 -M1x'2
xix2 - xrx?
In practice if the doublet being investigated is photographed near the center of the plate the deviations from
linearity turn out to be so small that there is no need for quadratic interpolation, and then
AM = A (Ax),
where the value of A may be computed with sufficient accuracy from the formula
A_ 2Mx (5)
,r1 + x2
In photographing doublets formed by ions of small masses (M < 13) the "base-lines" turn out to lie outside the
limits of the plate and therefore for the determination of the dependence of AM (Ax,x) in this case one must
use lines of an auxiliary spectrum (for example C4H3-C4H4-C4H5 or O-OH-OH2) registered on the same plate
as the doublet under investigation with the same ion-optical parameters of the apparatus. It is expedient to use
ions of the OHn group for the auxiliary spectrum since the mass of the 016 oxygen isotope is assumed to be equal
to 16, and the values for the mass of the hydrogen atom obtained by different methods' agree reasonably well
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Fig. 3: Photographs.of, mass=spectrographic doublets. a) , The triplet Nis?-N14H -C'lH3 (x40); b) the
Ld 14 16 16 14 12 14 12H
triplet NH 3-N HD-O H (x40); ?c) the triplet O N Hz-C. H4 (x40); d) the doublet N -G 2
(x60); e) the quartet 1/2 G12-..He4D-He4H2-D3,(x40).
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among themselves. The lines of the auxiliary mass-spectrum are brought to the photoplate by means of chang-
ing the strength of the magnetic field. In our experiments in view of the absence of a strict linearity in the
variation of the dispersion along the photoplate the magnetic field strength was varied so that the central "base
line" of the auxiliary spectrum would fall directly below the doublet under investigation at the center of,the
plate. In this case the relationship between AIM and the corresponding distance Ax between the doublet lines is
May
AM=Ax D
The influence of the leakage flux of the magnetic
analyzer was taken into account by introducing an ad-
ditional "magnetic (correction" coefficient into ex-
pression (6).
In all cases the distance between the lines of the
doublet was determined by means of measuring it ten
times on a comparator with an optical magnitification
of the image by a factor of 140 or 280. To exclude
subjective errors the measurement of Ax for any one
doublet was repeated.by three ;or,four operators, The
final value of Ax was obtained as the average value
of 10-12 doublets taken from 3-4 different photographs.
Experimental Results
The masses of the H, D. He4 and C1. isotopes were
found from the following relations:
where Mav is the average value of the masses of the ions constituting the doublet being investigated, and D Is
the dispersion constant determined from the relation
Mt
D =A
H
0+
~~-
-
6ga
q
8
6
4-
T,
He4
0-?2
4T-
in accordance with the data for the auxiliary spectrum (M1 is the mass corresponding to the central "base-line"
with an accuracy of 10-5).
For the determination of AM with an accuracy of 10-5 a.m.u. the value of May may be computed using
data on the masses of the isotopes of the light elements. As may be easily seen,for this it is sufficient to know
the value of May with an accuracy of 10'"4. The values available at present for the masses of the isotopes of the
lightest elements agree among themselves with the required limits of accuracy (10-4). For example.Tahle.1
gives values of May obtained by various authors for the doublet D-He4.
Investigator
Ewald [1]
Nier [4]
Ogata and Matsuda [3]
Li, Whaling et al. [6]
4.01662? 10.10 6
4.016674 ? 14.10-6
4.016680 t 12.10-6
4.016672 ? 14.10 -6
.z' la'"
986
Fig. 4., Mass defect M-A of the C12 isotope accord-
ing to the.data of: ' 1) Ewald, 1951 [1]; 2) Nier, 1951
[4]; 3) Li, Whaling et al., 1951 [6]; 4) Ogata and Mat-
suda, 1953 [3]; 5).Dzhelepov and Zyryanova, '152[7];
6) Mattauchand Bieri, 1954 [2]; 7) the present work.
is = 3 3 1 3 ~' .
C
Da-2
aA1t1A_D),
AM..D,a212
AM(C12H4D),
B=AM,(D2 =. He4),.
are respectively the mass differencesforithe Obublets
H2-D, Ds-1/2C'2, C'H4-O. D2-He4.
167
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TABLE 2
Mass Differences of the I)onhlets In 70-s a.111 u.
:,
.
/
Investigator
1)2 -110
9(,x2
1
1)~
i
Data of present work
1 .5118:1 I-0,001
.25.600 1:0,002
42,2981-0.007
30.388 10, 004
Ewald (1951) [1]
1.55.03 I.O, 0015
25.604: j 0,008
42,292.-4;0,012
36,371 1:0,012
Nier (1951) [4,5]
1.5519:1-0, 00172
25.(02-1-0,009
30,4274:0,008 *
Ogata and Matsuda(1953) [3]
1.54924.0.0008
25.603:10,000
42.301}_0,009
30, 4100,00(1
Whaling et al. (1951) [6)
Li
1.5494+0,0024
25.5901_0,008
42,3021:0,0113
30,372 )._0,019
,
Mattauch and Bieri(1954) [2]
1.5473?0,0076
25.60001-0,0047'.
42.3254?0,0052
36.4080?0, 00,38
") C o lima, N I e r and John son, l'hys. Rev. 84, 717 (1951).
TABLE 3
Masses of the H, D. [3e4 and CJ2 Isotopes
Investigator I M(H) I M(D) I M(Ho4)
M (C12)
Ewald (1951) [1]
1,008141?2
2,014732?4
4.003860?12
12.003807?11
Li, Whaling et al. (1951) [6]
1,008142?3
2,014735?6
4.003873 t15
12.003804?17
Ogata and Matsuda (1953) [3)
1.008145?2
2,014741?3
4.003879?9
12.003844?6
Nier (1951) [4,5]
1,008146?3 **
2,014778?8
4.003944?19
12.003842?4 **
Mattauch./and Bieri (1954) [2]
1,0081459+0,5
2.01474444} 0,9
4.0038797 ?1,6
12.003823?3,3
.Data of the present work
1.008142?1
2.014736_?2
4.003872?4
12.003820?5
*).Experimental errors are given in pa.m,u.
**) Co l I i n s, N i e r and Johnson, Phys. Rev. 84, 717 (1951).
The final results of the measurement AM of these doublets are given in Table 2. (iThe values for the
mass differences of. the ions are given in thousandths of a.m.u.). The calculated values of the masses of H, D,
He4 and C12 are given in Table 3. In the same tables are given corresponding data obtained in recent years
by other workers. .
Conclusion
The values of the isotopic masses of H and D. obtained in this work agree, sufficiently well with the data
obtained by Li, Whaling et al. [6] from nuclear reactions, and by Ewald [1] with a mass-spectrograph. The
deviations of our values of the masses of H and D from the data obtained by Ogata and Matsuda [3], by Mat-
tauch and Bieri [2], and by Nier and collaborators [ 4, 5 ] do not exceed the possible errors. The value for the
mass of the He4 isotope obtained by us agrees with the value obtained from nuclear reactions, and within the
limits of experimental error agrees with the data of Ewald, Nier,. Ogata and Matsuda. The values of the mass) of the C12 isotope as may be seen from Table 3 may be divided into two groups (Fig.. 4).. The values for the
mass of Cn obtained by. Nier [4, 5] and by Ogata and Matsuda [3], differ, from the corresponding data of Ewald
[1], Mattauch, Bieri [2], L,i, Whaling et al, [6] and Dzhelepov and Zyryanova [7] by about 3 x 10-5 a.m.u. with
an experimental error of less than 10-5 a.m.u.
The value obtained by us for the mass of C2 almost coincides with the data of Mattauch and Bieri and
within experimental error, agrees well with the values obtained by Li, Whaling et al. [6] and by Ewald [1]. Such
a good agreement of the values of the mass of C12 obtained by different independent methods indicates that
the data for the mass of 0 in this group of measurements are the more reliable ones.
168
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At the same time the existence of such a discrepancy
between the data of this group of measurements, and the
4
m
ONa data of Nier [4, 5] and Ogata and Matsuda [3] points to
OH ?
the presence of still investigated systematic errors due
most likely to the ion-optical peculiarities of the mass-
spectrographs.
The author consider it to be their duty to express
their gratitude to M. Ardenne for valuable advice and
and also to G.Jaeger. I. A.
for his interest in this work
,
Fig. 5. Spectrum for the determination of the disper-
Chukhin, and V. Roggenbuk for the help given by them
in adjusting the equipment.
Note: After the present article was completed a communication by Nier and collaborators [10] appeared giving
new preliminary values of masses obtained on a new mass-spectrometer with a maximum resolving power of
approximately 100,000: H = 1.0081439 f 5; D = 2.0147380 f 10; C12 = 12.0038174 t 18.
LITERATURE CITED
[1] H. Ewald, Z. Naturforsch, 6a, 293 (1951).
[2] J. Mattauch and R. Biers, Z. Naturforsch, 9a, 303 (1954).
[3] K. Ogata and H. Matsuda, Phys. Rev. 89, 27 (1953).
[4] A. Nier and T. Roberts, Phys. Rev. 81, 507 (1951).
(5] A. Nier, Phys. Rev. 81, 624 (1951)..
[6] C. Li, W. Whaling, W. Fowler and C. Lauritsen, Phys. Rev. 83, 512 (1951).
[7] B. S..Dzhelepov and L. N. Zyryai ova, Uspekhi Fiz. Nauk 48, 4651'(1952).
(8] Drummond, Phys. Rev. 97, 1004 (1955).
[9] A. H. Wapstra, Physica 21, 367. (1955).
[10] A. Nier et al., Bull. Amc.. Phys. Soc. No. 7, 18 (1955).
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 Declassified and Approved For Release 2013/04/03: CIA-RDP10-02196R000100090002-3
INVESTIGATION OF GAMMA-RAYS EMITTED BY NUCLEI OF CALCIUM,
NICKEL AND POTASSIUM ON CAPTURING THERMAL NEUTRONS
B. P. Adyasevich, L. V. Groshev, A. M. Demidov, V. N. Lutsenko
The energies and intensities of y -rays emitted by nuclei of calcium, nickel and potassium when they
capture thermal neutrons were measured by a magnetic spectrometer which analyzes the compton -electrons.
The y -ray spectra were studied in the energy interval 0.25 -12 Mev. The intensities of y -rays are expressed In
terms of the number of y -quanta emitted per 100 neutrons captured. Possible y -transition diagrams have been
constructed for Ca41, Nib9, Ni" and K40 nuclei.
The present work is a continuation of the investigation of y -rays emitted by nuclei on capturing thermal
neutrons which is being carried out with the RFT reactor of the Academy of Sciences of the USSR. The experi-
mental conditions, the method of measurement and the spectrometer have all been described before [1]. Below
results are given on the investigation of y -rays from the nuclei of calcium, nickel and potassium.
For the measurement of y -rays emitted on neutron capture by calcium of natural isotopic composition
a sample of CaF2 of 1.7 kg weight was used. In addition the y -spectrum was investigated for a sample enriched
in the Ca40 isotope. In this case the salt CaCO3 was used in the amount of 360 g. Before measurements were
made the samples were.specially purified from small chlorine impurities which, as was shown by preliminary
measurements, could make an appreciable contribution to the spectrum because of the large neutron absorption
cross section for chlorine.
TABLE 1
-
o s
?o b s
-~
[
s
y
y.
y .~
opu
u
U q
a
r' a
a
0
y d
o'
f~,
4
O
;.
E X
E
o
0
U o E
U o
4
4
z . a
-S 0.
Ca90
96.82
99.9
0.22
0.21
8.367?0.01
Ca42
0.64
-0.03
39.7
0.25
7.93?0.02
Ca93
0.129