SOVIET ATOMIC ENERGY VOL. 30, NO. 5

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• Document Creation Date: December 27, 2016
• Document Release Date: February 25, 2013
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• Publication Date: May 1, 1971
• Content Type: REPORT

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ILLEGIB
Rcissian Original VoL,.30, No. 5?May, 1971
Translation ,published November; 1971
SOVIET.
ATOMIC.
ENERGY
`ATOMHAR 3H,EPfVIR
(ATOMNAYA 'ENERGIYA)
TRANSLATED FROM RUSSIAN
CONSULTANTS BUREAU, NEW YORK,
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SOVIET
ATOMIC
ENERGY
Soviet Atomic Energy is a'cover-to-cover translation of Atomnaya
Energiya, a publication of the Academy of, Sciences of the USSR.
An.arrangement with .Mezhduriarodnaya Kn gar the Soviet book
export agency, makes available both advance copies of the Rus-
sian journal and original glossy photographs and artwork. This
serves to decrease the necessary'time lag between publication'
of the original and publication, of the translation and helps to im-,
prove the quality'of.the latter. The translation began with the first,
issue,-of the Russian journal.
Editorial Board of Atomnaya 'Energiya:
Editor: M.,D*Millionshchikov
Deputy Director
I.,V: ?Kurchatov Institute of Atomic Energy
Academy of Sciences of the USSR
Moscow, USSR
Associate Editors: N. A. Kolokol'tsov
N. -A: Vlasov
A:,I. Alikhanov ~
A. A.'Bochvar
N. A.' Dollezhal'
V. S. Fursov,
I. N.:Golovin
V. F. Kalinin.
A. K. Krasin
A. I. Leipunskii
V. V. Matveev
M. G. Meshcheryakov
P. N. Palei
V. B Shevchenko
D. L. Simonenko
V. I. Smirnov
A. P. Vinogradov
.A. P. Zefirov
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SOVIET ATOMIC ENERGY
A translation of Atomnaya Energiya
Translation published November, 1971
Volume 30, Number 5
May, 1971
CONTENTS
Increasing the Campaign of the First Unit of the Novo Voronezh Atomic Power Station
by Reducing the Power before Recharging the Fuel - F. Ya. Ovchinnikov,
L. I. Golubev, and S. F. Gaivoronskii ...............................
Monitoring the Fields of Energy Evolution by Reference to the Relative Efficiency of
the Control Rods - I. Ya. Emel'yanov, B. G. Dubovskii, Yu. V. Evdokimov,
L. V. Konstantinov, I. K. Pavlov, V. V. Postnikov, E. 1. Snitko, and G. A. Shasharin
The Method of Subgroups for Considering the Resonance Structure of Cross Sections
in Neutron Calculations - M. N. Nikolaev, A. A. Ignatov, N. V. Isaev,
and V. F. Khokhlov ...........' ...............................
High-Temperature Embrittlement of Neutron-Irradiated Steel Khl5N35V3T
- B. B. Voital', Sh. Sh. Ibragimov, V. N. Shemyakin, A. N. Vorob'ev,
L. A. Syshchikov, and A. G. Vakhtin ...............................
Radiation Damage of Beryllium during High-Temperature Irradiation
Z. I. Chechetkina, V. P. Gol'tsev, V. A. Kazakov, G. A. Sernyaev,
and V. G. Bazyukin ......................................... .
Radiolysis of Uranium Hexafluoride - V. A. Dmitrievskii and A. I. Migachev .........
Engl./Russ.
519 419
523 422
528 426
534 430
538 434
543 438
ABSTRACTS
Green's Function of the Neutron Transport Equation for a Moving Medium
- E. A. Garusov and Yu. V. Petrov ................................ 549 444
Stability of the Directional Distribution of Neutrons in Reactors with a Discrete Control
System - I. S. Postnikov and E. F. Sabaev ........................... 550 445
Hydrodynamics of Fissionable Materials. I. Acoustic Vibrations at Constant Neutron
Flux - V. M. Novikov .......... .............................. 551 446
Absorption of y-Radiation in Radiation-Chemical Reactors in Processes Carried Out
in Heterogeneous Agitated Systems - L. V. Popova, B. M. Terent'ev,
N. V. Kulikova, S. K. Dubnova, and A. Kh. Breger ...................... 552 447
The Use of High-Voltage Alternating Current for Industrial Radiation Processes
- L. V. Chepel'........ I ..................................... 554 448
A Method of Calculating the Basic Parameters of Radiation Installations in the
Irradiation of Stationary Systems with Accelerated Electrons - K. 1. Nikulin..... 554 448
The Monte Carlo Albedo Monoenergetic y-Rays Normal to Barriers of Various Media
- D. B. Pozdneev, N. V. Krasnoshchekov, and A. V. Pichugin ....... .... 555 449
Dependence of the Electronic Statistical Sum of Atoms on the Cutoff Parameters of
Energy Levels - A. A. Zaitsev and R. A. Kotomina ...................... 556 449
Effect of Pressure on Electronic Statistical Sums of Atoms at High Temperatures
- A. A. Zaitsev and R. A. Kotomina ................................ 557 450
Particle Trajectories in an Isochronous Cyclotron in the Presence of Acceleration. II
- Yu. K. Khokhlov ........................................... 558 451
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CONTENTS
Engl./Russ.
Critical Current in an Accelerating Waveguide with Radial Slits Provided in Discs
- A. K. Orlov..................................................
559
451
Magnetic System of a Superconducting 50 MeV Isochronous Proton Cyclotron
- L. A. Sarkisyan ............................................
560
452
LETTERS TO THE EDITOR
Physical Properties of Uranium Dodecaboride - V. V. Odintsov and Yu. B. Paderno ....
561
453
Asymptotic Behavior of a Neutron Pulse in a "Hard" Breeder System
- Yu. A. Platovskikh and I. V. Sergeev ......... . .............. ....
563
454
Scattering of ,14 MeV Neutrons by Iron - M. E. Gurtovoi, E. P. Kadkin, A. S. Kukhlenko,
B. E. Leshchenko, V. M. Neplyuev, G. Peto, and L. S. Sokolov ........
566
455
The Use of the Nuclear Reaction 018(p, a)N15 to Study the. Oxidation of Metals
- N. A. Skakun, A. P. Klyucharev, O. N. Kharkov, V. F. Zelenskii,
and V. S. Kulakov .................. ........................
569
456
Computer Simulation of Radioactive Decay Processes - G. G. Akalaev ............
572
459
Estimate of Emergency Doses at High-Power y-Facilities - E. D. Chistov,
I. F. Sprygaev, I. P. Korenkov, A. V. Terman, and A. V. Sedov ..............
575
460
Use of Semiconductor Detectors with a p-n Junction for the Dosimetry of X-Rays
and y-Radiation in the Low-Energy Range - V. K. Lyapidevskii
and Yu. B. Mandel'tsvaig ......................................
578
462
Proton Current for Equilibrium Phase Decrease along, a Linear Accelerator
- A. D. Vlasov ..............................................
581
464
Possibility of Accelerating Protons to Energies above E0 in an Isochronous Cyclotron
- L. A. Sarkisyan ...........................................
584
466
Turbulent Heating and Confinement of a Plasma in a Toroidal Trap of Multipole Type
B. A. Demidov and S. D. Fanchenko ...............................
587
468
NEWS
International Conference on Instrumentation in High-Energy Physics -I. A. Golutvin,
V. V. Vishnyakov, N. A. Toropkov, and Yu. A. Shcherbakov ................
590
471
Problems Encountered in Handling of Transuranium Elements - V. N. Kosyakov ......
595
474
Testing;of the "Stavrida"? Naval y-Facility ...............................
597
476
The "Betamicrometer" Radioisotopic Coating. Thickness Gauge -I. I. Kreindlin,
V. A. Novikov, and A. A. Pravikov ................................
600
478
Mobile y-Unit "Stimulator" - D. A. Kaushanskii ............................
602
479
An Atomic Energy Source for Electrical Cardiostimulators - E. B. Babskii,
E. E. Geronin, V. A. Kremnev, and G. M. Fradkin .. ..................
604
481
The Russian press date (podpisano k pechati) of this issue was 4/13/1971.
Publication therefore did not occur prior to this date, but must be assumed
to have taken place reasonably soon thereafter.
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INCREASING THE CAMPAIGN OF THE FIRST UNIT OF THE
NOVO VORONEZH ATOMIC POWER STATION BY REDUCING
THE POWER BEFORE RECHARGING THE FUEL
F. Ya. Ovchinnikov, L. I. Golubev, UDC 621.311.2:621.039
and S. F. Gaivoronskii
It is well known that reactors of the water-cooled, water-moderated type have a negative power effect
in relation to their reactivity (the Doppler effect). On reducing the power the degree of xenon poisoning is
also reduced. When the reserve of reactivity falls to zero at the end of a campaign, it may thus well be
possible to increase the depth of burn-up of the fuel and reduce the net cost of 1 kW/h of production by grad-
ually reducing the power of the reactor and thus releasing a certain amount of additional reactivity.
Under these conditions (spontaneous reduction in power), the critical state of the reactor is preserved
1) The increase in the breeding factor, following the reduction in power, arising from the Doppler ef-
fect (dpN), the reduction in the temperature of the heat carrier (dpT) and the reduction in xenon poisoning
(dPXe)
2) The reduction in the breeding factor arising from the burn-up of the fuel and the accumulation of
slags (dpbu).
Thus the overall balance of reactivity may be written in the following manner:
dP bu = dPN - dpT - dpxe? (1)
The reactivity effects associated with the changes in power, heat-carrier mean temperature, and op-
erating time of the reactor (amount of power developed) may in turn be written as follows:
dpN= -a(N) dN;
(2)
dpT= -fl(T)dT;
(3)
dPxe= -(N)dN;
(4)
dPbu
K (t) dW K (t) N (t) dt,
(5)
where the quantities a,/3, t, and K are usually known from calculations and experiments made when starting
the reactor.
TABLE 1. Composition of the Fourth Fuel
Charge in the First Unit of the Novo Voro-
nezh Atomic Power Station in Relation to
the Build-Up.of Slags in the Fuel Cassettes
2,0
1,5
0,714
Number of fuel cassettes with a build-up of
slag ysl, kg/ton U
54
9
3
5,0 1 it follows from this equality that
the neutron velocities are directed only in the direction of motion of the medium and are contained within
the cone 1 > p, ? fl - y-2. Thus, if y > 1, we have G = 0 for all z < zo, i.e., the neutrons are completely
entrained by the medium. It should be noted that the eigenfunctions that are found enable one to construct
not only the Green's function for an infinite medium (for which there is no need once it has been obtained
explicitly) but also to construct the solutions of various problems for a semi-infinite medium [1]. In par-
ticular, if the medium moves from a boundary in the albedo,problem and y > 1 for the incident neutrons,
all the eigenfunctions vanish for z < zo and this means the absence of reflected radiation.
1. K. Case, Ann. Phys., 9, 1 (1960).
2. E. A. Garusov, A. A. Kostritsa, and Yu. V. Petrov, At. Energ., 21, 128 (1966).
3. A. A. Kostritsa, At. Energ., 14, 218 (1963).
STABILITY OF THE DIRECTIONAL DISTRIBUTION OF NEUTRONS
IN REACTORS WITH A DISCRETE CONTROL SYSTEM
The stability of the directional distribution of neutrons in power reactors with.a discrete control sys-
tem is considered. The following control algorithm is proposed: a -full, instantaneous compensation for the
deviation of the power from its stationary value is effected at the end of each adjustment step (the step width
is assumed to be:constant). A similar problem was considered in [1] for the case of "pointwise" equations
of reactor kinetics. The present article is a generalization of the results of [1] to the case of the equations
of the diffusion approximation.
With these assumptions, the equations describing the dynamics of a reactcr with feedback derived from
the xenon contamination have the form
Acp -,- Lx -~,- R=O. rEQ;
7 1 =Px { 44, = ( x 1 . . . . . , xv).
(1)
(2)
where (p denotes the derivation of the neutron flux from its stationary value; A is a linear operator whose
spectrum is entirely located in the unbounded right semiplane; P is a v x v matrix; q, b, and x denote v x 1
column vectors; P denotes the transposed matrix; (p is a scalar; and R is the regulating force. According to
Translated from Atomnaya Energiya, No. 5, p.445, May, 1971.
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the control algorithm selected, it is assumed that the function R(r,t) is independent of t in the time inter-
vals (nT, (n + 1) T), where n = 0, 1, 2, . . . ; it is also assumed that the function changes at t = T, 2T, . . , nTR
(r, t) by an amount such that. cp (r, nT) = 0 for all rE Sl. It follows from Eq. (1) that the function R(r,t) in the
time interval (nT, (n + 1) T) is
It was shown with the second Lyapunov method. and with. the Yakubovich matrix-inequality technique
[2] that for small widths T of the adjustment step, the stability of the zeroth solution of the system of Eqs.
(1)-(3) is reduced to the determination of the conditions under which the smallest eigenvalue of the boundary
problem
%(r)=-lnflic ,T ~ (jm)b(joiE--N)-1 q;
(OE[O, a) ..
and A* is the operator conjugated to the operator A. The applicability of the stability condition was con-
sidered for the particular case in which A = A* in Eqs. (1)-(3) and in which the system coefficients are in-
dependent of r.
The critical width of the adjustment step for a one-dimensional model of a power reactor was cal-
culated as an example in which the feedback derived from the xenon contamination was taken into account.
It was shown that the critical width of the adjustment step amounts to several hours and increases consid-
erably when a control system for the average power is used.
LITERATURE CITED
1. O. B. Ronzhin and E. F. Sabaev, At. Energ., 24, 269 (1968).
2. V.A.Yakubovich, Avtomatika i Telemekhanika, 26, 577 (1965).
HYDRODYNAMICS OF FISSIONABLE MATERIALS
I. ACOUSTIC. VIBRATIONS AT CONSTANT NEUTRON FLUX .
A. theory of reactor criticality has been developed [1] in which oscillations in the density of the fuel
are excited in the reactor. The reciprocal effect of the neutron flux on the hydrodynamics of the fissionable
gas can be neglected, according to the underlying assumptions. This assumption would be valid when the
amplitude of the neutron flux is not too large. At the present time, the possibility of utilizing gas-fueled
reactors for, MHD generators is being discussed in the literature. In pulsed MHD generators of this type,
the neutron ux would have to reach levels of 1017 neutrons/cm2 ? sec. In such a case the effect of the
neutron flux on the hydrodynamics of the fissionable gas could no longer be ignored safely.* It is demon-
strated in the article that the extent of this effect can be assessed from the value of the dimensionless ratio
cY =)3 (gNH/cpc),where H is the reactor dimension; N is the neutron flux; /3, cp, c are respectively the vol-
ume expansion coefficient, specific heat, and the speed of sound traversing the fissionable material; q is the
*This could bring about reactor instability. The problem is discussed'from the standpoint by Goryachenko
[2]
Translated from Atomnaya Energiya, No. 5, p.446, May, 1971. Abstract submitted June 19, 1970.
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5 10
G1/W o
Fig. 1. Dispersion of acoustical oscil-
lations in heat removal through the
lateral surface of the channel, accord-
ing to Newtonian law.
rate of heat generated per unit mass due to unit neutron flux.
In the case of an ideal gas a = (y -1)gNH/c3, where y = cp/cv.
In the case of UF6, when H - 1 m, the critical neutron flux
(a (Ncr) = 1) would be 1015 to 1016 neutrons/cm2 ? sec. The.
law of dispersion of acoustic oscillations in a fissionable gas
subjected to a constant neutron flux of sufficiently great in-
tensity is derived for different heat conduction mechanisms.
The diagram shows the dispersion of acoustic oscillations
along the channel axis for the case of heat exchanged with the
laws according to Newtonian law.
It is clear from the diagram that the speed of sound in-
creases by a factor of f2 in the case of slow oscillations. The
logarithmic growth rate of the oscillations peaks at the fre-
quency w = cvo = (y - 1)qN/c2. Dispersion of acoustic oscilla-
tions at a high value of the thermal diffusivity is investigated
(for the case of radiative heat transfer). In that case, in the
limit as k -0, co - k2/3. It is demonstrated that there exists
a portion of the spectrum where vg and vph, the group velocity
and phase velocity of sound, satisfy the constraints
vga ?c; vph >> C and vgpvph z c=.
The resulting formulas can be used in order to obtain qualita-
tive assessments of the dispersion of sound in the gas flowing
through the reactor core.
LITERATURE CITED
1. V. M. Novikov, At. Energ., 27, 107 (1969); ibid., 30, 307 (1971).
2. V. D. Goryachenko, At. Energ., 24, 374 (1968).
ABSORPTION OF y-RADIATION IN RADIATION-CHEMICAL
REACTORS IN PROCESSES CARRIED OUT IN
HETEROGENEOUS AGITATED SYSTEMS
L. V. Popova, B. M. Terent'ev,
N. V. Kulikova, S. K. Dubnova,
and A. Kh. Breger
The Monte Carlo method was applied to calculations of the y-emission energy distribution of Co6o
in a radiation-chemical reactor with a heterogeneous medium under irradiation. A test-stand facility used
in the synthesis of dibutyl tin dibromide was employed to determine the energy of y-radiation absorbed in
the source, in the structural members, and in the reaction volume of the chemical reactor. The reaction
mixture constitutes a heterogeneous system consisting of metallic tin powder and liquid butyl bromide.
The dependence of the integrated absorbed dose in the reaction volume on the dimensions of the reac-
tion volume was investigated with the object of optimizing the performance parameters of the radiation-
chemical reactor. It was found that the highest efficiency of the equipment is achieved when the diameter
/height ratio of the radiation-chemical reactor is 1.2. Figure 1 shows how the absorbed dose rate varies
Translated from Atomnaya Energiya, No. 5, p.447, May, 1971. Abstract submitted March 9, 1970.
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P, rad/sec
Fig.,1. Relationship between absorbed
dose rate and diameter/height ratio (D
/fi) of. radiation-chemical reactor.
Fig.2. Dependence of absorbed dose
rate on distance in relaxation lengths (p
= 0.133 cm71) for different D/H values:
1) 2.5; 2) 1.07; 3) 0.65; 4) 0.31; 5) 0.13.
?) Engineering physics calculations; x)
Monte Carlo method.
with the diameter/height ratio (D/H) of the radiation-chemical reactor at two values of the weight ratio of
tin to butyl bromide, 1:1 and 1.10 (curves 1 and 2, respectively).
Calculations of the efficiency and distribution of the absorbed dose rate along. the radius of the radia-
tion-chemical reactor (with an axial irradiator) were carried out. Figure 2 shows the results of calcula-
tions carried out by two methods: by the Monte, Carlo method and by engineering physics calculations. The
maximum discrepancy in the results is 25%.
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THE USE OF HIGH-VOLTAGE ALTERNATING CURRENT
FOR INDUSTRIAL RADIATION PROCESSES
L. V. Chepel'* UDC 541.15:621.3.038.624
The use of electron accelerators, primarily those with a constant electron energy in the 300-500 keV
range, has recently increased in industrial processes.
However, it may be shown that for industrial accelerators the energy of the emissions generally need
not be constant in the given range. On the one hand,.every softening of the accelerator's energy spectrum
increases the fraction of the beam energy absorbed at the output window; on the other hand, because of the
specific nature of the electrons' interaction with the substance, the uniformity of the dose absorbed in the
surface layers of the irradiated material is improved.
Therefore, in principle optimal spectral shapes and conditions of irradiation may be found for which,
in the final analysis, the efficiency of the electron beam, for a given degree of uniformity of irradiation of
a material, will be comparable to the efficiency of a monoenergetic beam.
In this connection it is interesting to consider the possibility of using a high voltage alternating cur-
rent to feed the accelerator tubes of industrial radiation equipment. Realization of this possibility in the
300-750 keV energy range will simplify the construction of accelerator equipment to a considerable extent
and increase reliability through the use of either high voltage power supplies containing no rectifiers or
condensers, or electrical transmission lines at a corresponding voltage.
In the article the advantages of using an electron beam with a sinusoidal energy spectrum are evalu-
ated analytically. The corresponding integral curves are obtained by approximating the distribution curve
of the deep dose by a sine wave segment. From these curves the energy efficiency of the electron beam
and the degree of depth nonuniformity of the dose in the material were determined for irradiation from two
sides.
The method of sweeping the sinusoidal-spectrum electron beam by modulating the deviating magnetic
field was proposed. The shape of the modulating magnetic field sweep curve and the relationship of its fre-
quency to the electron kinetic energy were found.
A METHOD OF CALCULATING THE BASIC PARAMETERS OF
RADIATION INSTALLATIONS IN THE IRRADIATION OF
STATIONARY SYSTEMS WITH ACCELERATED ELECTRONS
K. I. N i ku l i nt UDC 621.039.83:541.15
In calculating the parameters of radiation installations [1] used for carrying out processes in station-
ary systems (the cross-linking of polyethylene, the vulcanization of industrial rubber products, etc.), using
accelerated electrons, one of the most important problems is to make sure that the absorbed doses are uni-
formly distributed throughout the volume of the irradiated material.
The initial data for calculating the parameters of radiation installations are the following: the dimen-
sions and shape of the object, the composition and density of the material, the magnitude of the absorbed
*Translated from Atomnaya Energiya, No. 5, p.448, May, 1971. Original article submitted May 28,
1970; revision submitted October 2, 1970.
tTranslated from Atomnaya Energiya, No. 5, pp. 448-449, May, 1971. Original article submitted July
10, 1970.
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dose, the allowable nonuniformity of the field of absorbed doses in the irradiated object, and the productivity
of the installation.
Investigations include the solution of the following problems: choosing the method of equalizing the
field of absorbed doses in the object, calculating the field of absorbed dose rates in the object; the nonuni-
formity of the field of absorbed doses in the material, and the conditions ensuring a prescribed uniformity
of irradiation; calculating the energy of the accelerated electrons, energetic efficiencies, the power of the
electron source, the velocity of motion of the material in the irradiation zone, and the number of accelera-
tors needed for a prescribed installation productivity. The first systematic investigations along this line
were described in [1-4].
The present study gives a method for calculating the magnitudes of absorbed doses throughout the
thickness of the material when it is irradiated with a wide beam of monoenergetic electrons. It describes
a method for calculating the nonuniformity of the field of absorbed doses throughout the thickness of the
material. It proposes irradiation methods which ensure that the absorbed doses will be uniform to within
:L5%. It gives equations which can be used to determine the efficiency of utilization of the electronic radia-
tion in various methods of equalizing the magnitude of the absorbed dose throughout the thickness of the
material. It describes a method for calculating the basic parameters of radiation installations inthe irra-
diation of stationary systems with a wide beam of monoenergetic electrons.
1. A. Kh. Breger et al., Fundamentals'of Radiochemical Apparatus Design [in Russian], Atomizdat, Mos-
cow (1967).
2. , Yu. S. Ryabukhin et al., At. Energ., 19, 535 (1965).
3. L. W. Chappell et al., in: Electron Accelerators [Russian translation], Atomizdat, Moscow (1966), p.
399.
4. K. I. Nikulin and G. A. Obraztsov, At. Energ., 23, 50 (1967).
THE MONTE CARLO ALBEDO MONOENERGETIC y-RAYS
NORMAL TO BARRIERS OF VARIOUS MEDIA
D. B. Pozdneev, N. V..Krasnoshchekov, UDC 539.122.04
and A. V. Pichugin
The Monte Carlo method is used for calculating, with a unidirectional source, the spectral-angular .
and integral characteristics of y-quanta back-scattered from various media. The study investigates the
case of the normal incidence of primary y-quanta with an energy E0 of 0.145, 0.279, 0.511, 0.662, 1.0, and
1.25 MeV on barriers made of Be, C, Al, Fe, and Sn, whose thickness in the normal direction measured 0.1,
0.2, 0.4, 0.6, 1.0, and 2.5 times the free-path length of the primary y-quanta. By means of a scintillation
y -spectrometer an experiment was conducted to investigate the spectral-angular distribution of the reflected
"narrow-beam" y-radiation of Cs137 from these media.
An analysis of the resulting data concerning the differential current numerical albedo for the above-
mentioned situations showed that its variation as .a function of the thickness may be described, with an ac-
curacy of ?10%, by the formula
I(0) (x)=-4(0) (oo) [1_e_a(O)Y1,
Translated from Atomnaya Energiya, No. 5, p.449, May, 1971. Original article submitted July 10,
1970; revision submitted October 21, 1970.
555
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where A(0) (x) is the differential numerical current albedo for a barrier of thickness x times the free-path
length; A (0) (oo) is the same for the case of a semi-infinite scatterer; a (0) is an empirical quantity.
A formula that may be recommended for finding A(0) (ao) is
A(? (oo)_(a-b cos 0) cos 0, (2)
where a and b are empirical quantities depending on E0 and Z; 0 is the exit angle of the photons, measured
from the normal.
The study gives recommendations for fi:iding the quantities appearing in formulas (1) and (2).
The variation of the integral numerical current albedo as a function of the thickness can be expressed,
to within f5-10%0, by the empirical formula
where A(-) is the integral-numerical albedo for a semi-infinite scatterer and a is an empirical quantity.
DEPENDENCE OF THE ELECTRONIC STATISTICAL SUM OF ATOMS
ON THE CUTOFF PARAMETER OF ENERGY LEVELS
The dependence of the electronic statistical sum of the atom (of a monatomic ion) on the cutoff para-
meter of the energy levels (understood here as DE, the maximum energy of an electron present in the elec-
tron in the bound state) was studied. The problem of approximate treatment of the contribution made by
the highest energy levels to the electronic portion of the statistical sum is also discussed in the article.
The assumption entertained in all the calculations is that the highest energy levels of the electron in the
atom are similar to the energy levels of a charged particle moving through a central coulomb field. For
such a field, the electron distribution is given by the formula
do,, =BE-5/2 dE, (1)
where dc OK is the number of quantum states of an electron with energy from E to E + dE, and B is a constant.
For convenience in the computations, the concept of the Coulomb statistical sum is introduced, and is under-
stood as the electronic statistical sum of the atom. All the electron levels are assumed to be Coulomb levels in
this sum. On the basis of Eq. (1), we proceed to find the explicit dependence of In QK on AE:
in Qa (AE) =1n Q,, +cp (AE), (2)
where In QI{ is a part of In QK(AE) which is independent of DE, and
q.(tE)-310%2exp(- )[32 2 AE3/2.f .-DE-1/2-(kT)2AEf/2+... ]:.
The dependence of the chemical potential I on DE and T is obtained in the form
I = I1?f (AE).
5
I1=Io (t+ 39o-f byu+4.745go +
...)
j (AE) _ W (AE, I1)
3/1/21= 5/2 (1+5.833y1+67.376 1 , /
yit. ?)~ kT W(AE1)
Yo ( 270 )' 1=(2102;
Translated from Atomnaya Energiya, No. 5, pp. 449-450, May, 1971. Original article submitted July 1,
revision submitted September 23, 1970.
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Io is the ionization potential of the atom; k is the Boltzmann constant. In order to find the dependence of
the true electronic statistical sum of the atom Q(AE) on DE, we break up the entire energy interval of the
electron in the atom into two parts such that levels lying above a certain level designated AE, may be
treated as coulomb levels. On the basis of Eq. (2), it is shown that the formula
will be valid for In Q(DE). The lower energy levels are known for most atoms, to sufficient accuracy; these
can be used to compute In Q(DE1) by direct summation. Accordingly, Eq. (5) can be used to compute the
electronic statistical sum of the atom for any cutoff parameter AE however small, provided the lowermost
energy levels of the atom and the ionization potential of the atom are known. The article cites the results
of computations of the electronic statistical sum of indium atoms and lead atoms, using the method pro-
posed, for temperatures up to 14,000?K. The data are compared to values of Qel of those atoms reported
in the literature.
EFFECT OF PRESSURE ON ELECTRONIC STATISTICAL
SUMS OF ATOMS AT HIGH TEMPERATURES
A. A. Zaitsev and R. A. Kotomina UDC 539.183.3
An approximate method is proposed for limiting the electronic energy levels of an atom present in a
weakly ionized plasma at high temperatures.' The atom is treated as a system consisting of a nucleus and
the electron gas surrounding the nucleus. The effect of the remaining atoms and ions surrounding the atom
in question and perturbing its outer electronic levels reduces to the fact that the electron gas is acted upon
by a certain external pressure of p equal to the total pressure exerted by the gas. That distance DE from
the nucleus at which the pressure exerted by the electron gas becomes equal to the external pressure p is
represented by the radius of the sphere confining the atom in question.
Assuming that the higher-lying energy levels of any atom are Coulomb levels, and applying to the elec-
tron gas the familiar thermodynamic relations
/
P=-1 7,,v,-) T;
we can obtain a formula linking the cutoff parameter AE of the energy levels and the pressure and tempera-
ture of the monatomic gas in the form
3[2 _ 1 l
P V2ntz3In(1+exp\4kTJ/JAE3/2' (2)
where m is the mass of the electron; k is Boltzmann's constant; J is the absolute value of the chemical po-
tential (the dependence of J on T, AE, and on the ionization potential of the atom J0 is obtained).
For practical applications, formula (2) is applied through the expansion of ln[1 + exp((E -J/kT)] in
a series in powers of ((E/kT), so that Eq. (2) can be transformed to the form
4E3/2 exp (
, kT 1,687 104kT exp
where (kT), J, and L E are expressed in electron-volts; and p is expressed in atmospheres. AE is calculated
from specified p and T by a graphical method.
The effective charge does not appear in Eqs. (2), so that ambiguities attendant upon the definition of
that parameter can be avoided, and Eqs. (2) can be applied to monatomic ions as well with no modifications.
Translated from Atomnaya Energiya, No. 5, p. 450, May, 1971. Original article submitted July 1,
1970; revision submitted September 23, 1970.
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The example of the lead atom is used over a broad range of temperatures and pressures in this meth-
od for calculating the cutoff parameter AE, and comparing it to methods developed elsewhere in the litera-
ture.
The procedure outlined is applied to calculations of the cutoff parai Teter of the energy levels of the
uranium atom for temperatures to 10,000?K and pressures from 0.01 to 100 atm. These data then provide
a basis for obtaining values of the natural logarithm of the electronic statistical sum and the reduced ther-
modynamic potential of monatomic uranium in theAdeal gas state, for the indicated range of temperatures
and pressures. It is pointed out that In Qel,of uranium obeys a marked pressure dependence at T = 10,000?K.
PARTICLE TRAJECTORIES IN AN ISOCHRONOUS
CYCLOTRON IN THE PRESENCE OF ACCELERATION. II
Yu. K. Khokhlov UDC 621.384.611
Time and spatial oscillations of a particle moving in the median plane of a cyclotron are discussed
in the article. "Ideal" curves, i.e., trajectories rotating uniformly [1], are used as the coordinate curves.
On each portion of the trajectory outside the slits, the position of a particle possessing a momentum p is
reckoned from the ideal trajectory belonging to the same momentum p according to the formulas
i(d)=t(w)-did (w); z.(S)=[R (q)-R.id ((p))cos''V((p)-
Here t is the time; R, cp are polar eoordinates;,9 =,9(cp) is the generalized azimuth; tan t((p) = Re1((p)/Re(cp);
Re (cp) is the equilibrium (closed) orbit. The linearized equations of motion outside the slits and the boundary
conditions on the slits acquire the form
tlt9 X (1')=G (0) X (0); AX (SJ)= -Py'Eir (ei) X (15i)
Here G69), I' (,9) are three-row matrices all of whose elements are defined on the corresponding equilibrium
orbit Re(cp); X69) is a vector with the components T(,9), x((9), x'(,9); .X(,9) is the jump increment of the vector
X(,9) upon traversing the j-th slit; e = co '[dAW(t)/dt]t_t is the "varying acceleration" parameter; We is the
frequency of revolution of the particle; OW(t) is the increment in the kinetic energy W on the slit at
time t.
The system (1) is solved in terms of generalized amplitudes and phases in Floquet theory extended
to the case of quasiperiodic motion. The resulting formulas describe the behavior of the vector X(2irn) as
a function of the number of revolutions n when the azimuth of observation is fixed. The vector X(27rn) moves
with a frequency y = 27r(vx - 1), where vx is the frequency of radial betatron oscillations over the surface
of a three-dimensional ellipsoid whose parameters in turn vary smoothly with increasing n.
In the case of a cyclotron of the type described in [2] with two dees, all the effects related to E are
divided into proportional parts I el I + I EII I and I EI I - I EII I (I, II are numbers assigned to the slits in the
semirevolution 0 < cp < 7r). Under rigorously isochronous acceleration conditions we have I EI I = I ell I , i.e.,
effects of the second kind are absent. Effects of the first kind are manifested specifically in an appreciable
inclination of the axes of the ellipsoid, the shift of the center of the ellipsoid relative to the origin of co-
ordinates, "smearing" of the boundaries of the bunch, and of the energy distribution. The sufficient condi-
tion for energy separation of bunches of adjacent n values takes the form I Wn(Xno) - Wn(0) I a [1].
In this paper an attempt is made to determine the asymptotic behavior of fission chamber responses .
m(t) when there are no discrete decay constants. We shall consider that the system is homogeneous and
"hard" enough that there,are no thermal neutrons in it. It is shown in [1] that if a Laplacian transforma-
tion is applied to the equation
a1 + Ln = X (v) vE fn (v') v' dv'-+ S (v) & (l),
0
then the Laplacian transformation m(s);of the.fission chamber responses may be written in the form
m (s) = 1 k k%S(s)
In Eqs. (1) and (2), n = n(v, t) is the neutron density;- L. is- an operator which describes neutron moderation
and accounts for leakage:in theform DB2; X (v) is the fission spectrum; S(v) is the spectrum of the pulse
source; and
m'= vEfnv dv; n= ne-Et dt;
e Jp
ao ~
ks= S vEfnsvdv; kX= f VZfnXVdv,
where ns and nX are solutions to the equations J
(s+L)ns=S, (s+L)nx=X.
Characteristic values (poles of m(s)) are determined from the equation kX (s) = 1. When s = -a, m(s)
has a branch point. In our case m(s) is monotonic on-the real axis when--a < s < on, and so there is only
one characteristic value (or none). The equation kX (-a) = 1 determines the boundary size of the system at
which the discrete characteristic number reaches the bounds of-the continuous spectrum. The quantity
k X (-a) is.equal to the. effective neutron multiplication coefficient in a: system without 1/v absorption.
In Wigner's approximation
Ln=vEn- 2:vb ! Es (v') v -bdv'; S= -1.
If we consider that vE = vDB2 + vEs + a and vE f = fl, where D, Es, a, and'a are constant, then for a system
boundary- size corresponding to B2 the following: equation may be found:
Translated from Atomnaya Energiya, No. 5, pp. 454-455, May, 1971. Original letter submitted- June
21, 1970.
? 1971 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York,
N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without
permission of the publisher. A copy of this article is available from the publisher for $15.00.
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DBo=Sv I xdu+~2.,
0
When B > Bo, there are no discrete characteristic numbers.
Suppose that both sources are monoenergetic: X(v) = 6(v -vo), S(v) = 6(v -vs). Then in Wigner's
approximation we obtain
Ak=1-k%(-a)=1- PV
Eo - E2VS(Ps 1)
Eo=Es+DB2; P=2DB2
Eo
EO
If vs >> vo and 1 < p < 2, then for small values of w corresponding to large t, we obtain the approxima-
() 1 PvEBr (6+2) r (1-P)
M to w 11 P-1 ' c Eouor (8 -P+2)
Ak+c vo /
in which Ak ? 0 when there are no discrete characteristic numbers, and Ak < 0 when there are. A discrete
characteristic value near the boundary of the continuous spectrum is determined by the expression
tuo -a-k~Ak1 P-1
VO U0E0 -( C
(7)
,which is analogous to the formula for the reciprocal numbers when the subcriticality is great (A close to a).
The residue of m(w) when co = w0, i.e., the amplitude of the exponential part of m(t), approaches zero when
A -a. When there are no discrete characteristic values, the inverse Laplacian transformation of expression
(6) yields a real integral, and the asymptotic behavior of m(t) is represented as follows:
-P +A Aot (8)
nz (t) a-at (t+A t l , )
where AO = Eowo.
If p > 2, then w0 -Ak, and m(t) is also represented by expression (8) when there are no discrete char-
acteristic numbers, but the second exponent is proportional to -Akt. Equation (8) also describes m(t) when
X (v) = vze-av2 (actual fission spectrum). The limits show that, for not very large t, large p, and Ak, the
second term of expression (8) will be important, i.e.,
e-(a+AO) t (g)
which corresponds to nearly exponential behavior of m(t) outside the continuous spectrum with a time con-,
stant of a + A0. For large times (t -:- o) the first term of Eq. (8) will predominate, and m(t) - t-pe-at, where
the exponent no longer depends on the subcriticality. Thus, outside the continuous spectrum three types of
detector indications are possible, depending on the system parameters and time.
1) There is no definite exponent when both terms of m(t) are about equally important at the time of
observation. In this case m(t) will be a concave function.
2) The behavior of m(t) is nearly exponential, with a time constant a + AO which depends on the. sub-
criticality. This time constant may be considered a characteristic of the system, even when there is a dis-
crete time constant. The quantity a + AO is related to the effective -multiplication constant in the same way
a discrete characteristic number is [see Eq. (7)].
3) The behavior of m(t) is nearly exponential, with a time constant a, which does not depend on the
subcriticality.
At least the first two types of behavior in m(t) are observed in experiments with "hard" installations
[1, 21. For other energy functions of the cross section, the asymptotic behavior of m(t) remains the same,
but the exponents change in the preexponential factors.
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svoov kx((,) ) - Eo v0pv + Eo (8+1) (vo+w)6-Pw 8+P-2 2F1 (8-P+2, 6+1; 8+2;
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LITERATURE CITED,
1. F. Storrer, Pulsed Neutron Research, Vol.2, IAEA, Vienna-(1965).
2. W. Patterson et al., Pulsed Neutron Research, Vol. 2, IAEA, Vienna (1965).
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SCATTERING OF 14 MeV NEUTRONS BY IRON
M. E. Gurtovoi, E. P. Kadkin,
A. S. Kukhlenko, B. E. Leshchenko,
V. M. Neplyuev, G. Peto,
and L. S. Sokolov
The characteristics of excited states of the Fe56 nucleus have been comparatively closely studied.
The first 2+state (0.85 MeV) and the 3+ state (4.51 MeV) are strongly excited in inelastic scattering of elec-
trons, protons, dueterons, and a-particles [1-5]. These states are satisfactorily represented by the collec-
tive (vibration) model in cross-section calculations by the distorted-wave method, and in the formalism of
linked channels. The deformation parameters al are in the following ranges: 02 0.18 + 0.29; 03 = 0.10
+ Q.23. It was found that f3l depends very little on the type and energy of the bombarding particles, but does
depend on the method of analysis. The scatter in the values of fl, is fairly great, and this introduces in-
determinacy into the interpretation of the results.
The aim of our present work was to study inelastic scattering of neutrons by Fe56 with excitation of
the level at 0.85 MeV, to determine whether or not this obeys the laws following from the collective nature
of the 0.85 MeV state, and to estimate as accurately as possible the value of 02 obtained from an analysis
of the data by the method of linked channels.
We measured the differential cross sections of elastic and inelastic scattering of 14 MeV neutrons
by iron nuclei. We used a time-of-flight spectrometer with a pulsed neutron source. Pulses of deuterons
were obtained from a low-voltage neutron generator by grouping the deuteron beam before acceleration [6].
The pulse duration was about 1 msec, the repetition frequency 7 MHz, and the pulse current about 3 mA.
The neutrons were generated by the reaction T(d, n)4He with a standard tritium-titanium target. With a
flight base of 10 m, the energy resolution of the spectrometer was 3% (about 450 keV). Time marks for
the convertor were obtained from an induction electrode placed near the target. The neutron detector was
a plastic scintillator, 80 x 80 mm in size, connected to an FEU-36 photomultiplier., The registration thresh-
old was 7.5 MeV. We used annular geometry. The mean diameter of the iron toroid was 33 cm, and the
diameter of the cross section was 3 cm.
The measurement results are plotted in Fig. 1 [1], together with the results of other authors [7-9].
The cross sections were analyzed by the method of linked channels [10, 11] on the assumption that
the 0.85 MeV level is collective. We used an optical potential with surface absorption. Calculations were
performed with and without the spin-orbit term.
TABLE 1. Optical Potential Parameters used in Calculation
Num-
ULS,
ber of !V,MeV jW,MeV
MeV
a,F
ro,F
aW,F
rw,F
82
curve
1 43.9 4.35
0
0.515
1.25
0.7
1.2
0.18
2 43.9 2.7
9
0.39
1.25
0.7
1.25
0.23
Note: The parameters V, to and a characterize the depth, radius and blurring
of the boundary for the real part of the potential; the parameters W, rw, and
aW represent the imaginary part; and the parameter ULS denotes the depth of
the spin-orbital potential.
Translated from Atomnaya nergiya, No. 5, pp.455-456, May, 1971. Original letter submitted Septem-
ber 22, 1970.
o 1971 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York,
N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without
permission of the publisher. A copy of this article is available from the publisher for $15.00.
566
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Fig. 1. Differential cross sections of
elastic and inelastic scattering of 14
MeV neutrons by iron. A) Elastic
scattering; 0) inelastic scattering
with excitation of 0.85 MeV level; ?)
[71; x) [8]; [1) [9].
U 20 40 50 80 100 120 140 160 180
0 (c. m. s. ), deg
As initial potential parameters we used parameters taken from the paper of Rosen et al. [12]. Then
the calculated cross sections were fitted to the experimental values by varying all the parameters except
ro and aw. The quadrupole deformation a2 was taken to 0.23 and was also varied during fitting. The optimal
theoretical curves are plotted in Fig. 1, and the corresponding values of the optimal potential parameters
are listed in Table 1.
As we see from Fig. 1, the collective (vibrational) model gives a satisfactory description of the cross
sections for elastic and inelastic scattering of neutrons by Fe56 nuclei. The parameter /32, found as a result
of our analysis, is equal to 0.21 f 0.03, which-agrees with the corresponding values of /32 in [1-5].
LITERATURE CITED
1. R. Peterson, Ann. Phys. (USA), 53, 40 (1969).
2. H. Tjin a Djie et al., Nucl. Phys., A106, 85 (1968).
3. A. Majumder and H. Sen Gupta, Nucl. Phys., A.118, 151 (1968).
4. C. Fulmer et al., Phys. Rev., 165, 1218 (1968).
5. S. Fulling and G. Satchler, Nucl. Phys., A111, 81 (1968).
6: M. E. Gurtovoi et al., Pribory i Tekhnika tksperimenta, No. 4, 24 (1970).
7. J. Coon et al., Phys. Rev., 111, 250 (1958).
8. J. Anderson et al., Phys. Rev., 110, 160 (1958).
9. V. Jacques, C. R. Acad. Sci., 267, B733 (1968).
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 Declassified in Part - Sanitized Copy Approved for Release 2013/02/25: CIA-RDP10-02196R000300080004-0
10. T. Tamura, Rev. Mod. Phys., 37, 679 (1965).
11. E. V. Korbetskii, 0. F. Nemets, and L. S. Sokolov, Abstracts of Reports to Twentieth Yearly Conference
on Nuclear Spectroscopy and the Structure of the Atomic Nucleus [in Russian], Part 2, Nauka, Lenin-
grad (1970), p.140.
12. L. Rosen et al., Ann. Phys., 34, 96 (1965).
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THE USE OF THE NUCLEAR REACTION O18(p, a)N15
TO STUDY THE OXIDATION OF METALS
N. A. Skakun, A. P. Klyucharev,
0. N. Kharkov, V. F. Zelenskii,
and V. S. Kulakov
UDC 620.197.5.539.17
Data on nuclear reactions have recently been used for the study of the processes that occur in the
surface layers during the oxidation of metals and alloys [1, 2]. Recording the reaction products during ir-
radiation of the investigated material with charged particles permits a determination of the content of the
element to be studied simultaneously with a localization of it with respect to depth without destruction of
the sample.
The essence of one of the methods proposed in this -work is the recording and analysis of the energy
spectrum of a-particles from the reaction O18(p, a)N15 in bombardment of a sample containing the isotope
018 with a beam of monoenergetic protons. If the reaction takes place at some depth x, then the a-particle
energy E. depends on the energy losses of the proton that has reached a,depth x and the energy lost by the
a-particle during its escape from the sample. With an accuracy determined by the energy resolution, the
energy Ea indicates the depth at which the reaction takes place. The yield of a-particles Na (Ea) is pro-
portional to the product of the concentration C(x) of the isotope 018 by the cross section of the reaction at
a proton energy Ep(x). Thus, the energy spectrum of a-particles permits establishment of the distribution
of oxygen according to depth of the sample, if the stopping power of the investigated substance for protons
and a-particles and the cross section a (Ep) of the nuclear reaction are known.
a 15
a
O 5
U
-
-
--.~_
i
O
?oooo~?oo?
oo?op
600 650
Energy, - keV
Fig. 1
C
u 0,4
C
0
O ^
~o 9 00
8
9,2 9,4 0,6 48 1,0
Depth, W
Fig. 1. Excitation functions of the reaction O18(p,a)N15 at an angle els = 150?.
Fig.2. Distribution of oxygen in samples of zirconium oxidized for 30 (0), 50 (0), and 120
(0) min. The corresponding spectra ofa-particles at EP = 730 keV are shown in the in-
sert. Energy scale 16.7 keV/channel.
Translated from Atomnaya Energiya, No. 5, pp.456-458, May, 1971. Original letter submitted July
14, 1970.
O 1971 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York,
N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without
permission of the publisher. A copy of this article is available from the publisher for $15.00.
N
a
103
O
O
^
O
1.103
a
O
^
O
O^
.0
0
20 30 40 50
Channel number
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A sample with a constant concentration of the isotope 018 with respect to depth (standard) permits a
determination of the profile of the oxygen concentration in the investigated sample by comparison of the a-
particle spectra from the sample studied and the standard. In this case the value of C(x) is determined ac-
cording to the function
Na (Ea)st - Cst
where Cst = const is the concentration of 018 in the standard; Na (Ea) and Na (Ea)st are determined experi-
mentally. In this case the dependence a[Ep(x)] is considered automatically, if the energy losses of protons
(ep) and a-particles (Ca) per unit path in the investigated sample and standard are the same. The values
of ep and ea were calculated using the data of [3].
The excitation function of the reaction 018(p, a)N15 was measured experimentally in the interval of
proton energies 500-730 keV (Fig. 1). A. target Ta2O51 produced by anodic oxidation of tantalum in an elec-
trolyte enriched with the isotope 018, was used. The source of accelerated protons was a4 MeV electro-
static generator from the Physicotechnical Institute of the Academy of Sciences of the Ukrainian SSR. a
Particles were recorded with a .silicon surface-barrier counter at an angle 8 = 150? and were analyzed ac-
cording to energy with an AI-100-1 analyzer.
In the interval of the energy Ep indicated above, only one group of a-particles, corresponding to a
value of Q = 3.97 MeV (ground state of the N15 nucleus) is emitted in the reaction 018(p, a)N15 [4]. Elastical-
ly scattered protons were absorbed with a mylar film 1.16 mg/cm2 thick, placed directly in front of the
detector. The surfaces of the irradiated targets were established perpendicular to the axis of the proton
beam.
The investigated targets had dimensions 20 x 20 x 0.5 mm and represented samples of iodide zir-
conium, remelted by an electron beam method, oxidized at 500?C and a pressure of 1 atm in water vapors
containing 69% 018. A sample of the alloy Zr - 2% Nb, oxidized for a long time under the same conditions,,
was used as the standard. A comparison of the a-particle spectrum from a standard obtained at Ep = 600
keV with the excitation function in the region of Ep 600 keV, where the cross section v(Ep) obeys an ex-
ponential law, indicated that the concentration Cst(x) remains practically constant downto a depth of -1.2 p.
It has been suggested that Ep and ea in the oxide film formed on a sample of Zr - 2% Nb do not differ
significantly from the corresponding energy losses in oxidized zirconium.
The calculated stopping powers of zirconium oxide for protons and a-particles are equal to ep = 83
keV/p at Ep = 730 keV and ea = 310 keV/p at Ea = 3.38 Me.V. Using these data and the procedure outlined
earlier for comparison of the spectra of a-particles fromthe investigated samples and the standard, ir-
radiated with protons with an energy Ep = 730 keV, we can obtain profiles of the distribution of oxygen in
oxidized samples of zirconium (Fig. 2). The error in the determination of the relative concentration, related
to the statistical errors of count, is an average of no more than 3%. From the curves of Fig. 2 we estimated
the diffusion coefficient of oxygen in zirconium, the value of which was -7.5. 10-14 cm2/sec. The resolving
power of the metal with respect to depth, determined by the experimental energy distribution of the a-par-
ticles, is equal to -0.2 p. In the presence of a standard, the time expenditures necessary for determining
the profile of the oxygen concentration in each sample -(including the time for irradiation) did not exceed
10-15 min.
The possibility of conducting analogous investigations using the narrow resonance of the reaction
O18(p, a)N15 at Ep = 639 keV (see Fig. 1) was also considered. Figure 3 presents the curve of the resonance
yield of a-particles, obtained as a result of irradiation with monoenergetic protons in the interval 615-650
keV of a sample of tantalum, subjected to anodic oxidation in an electrolyte enriched with the isotope 018.
In this case, for each fixed value of Ep, we recorded the summary yield of a-particles, from which the con-
tribution of the a-particles due to the nonresonance cross section of the reaction was deducted. The thick-
ness of the oxide film was determined from the curve obtained in this way with the aid of the expression
r2=rIe5 4-r+0. (2)
where r is the width of the curve of the resonance yield at half the height (13.6 keV); rres is the natural
width of the resonance (2.1 keV); ri is the energetic inhomogeneity of protons in the beam (1.7 keV in our
case); rt is the thickness of the target (keV).
Na (E.) = C (z)
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0510 520 530 540 550 50 80 100 20
Energy, keV Channel No.
Fig. 3 Fig. 4
Fig. 3. Resonance yield of a-particles from a target of Ta2O5 1530 A thick.
Fig.4. Spectra of a-particles obtained from a sample of zirconium oxi-
dized for 120 min.
The stopping power of tantalum oxide Ta2O5 for protons in the energy interval used is 98 keV/?. Con-
sequently, the value Ft = 13.2 keV, obtained from function (2), corresponds to a thickness of the oxide film
of 1300 A.
The process of anodic oxidation was analogous to that outlined in [5]. The sample of tantalum was
oxidized at a voltage of 100 V in a 1% solution of KCl in water with enrichment with 12 atomic % 018.
In accord with the results of [5] it was assumed that the oxide has a stoichiometric composition cor-
responding to the chemical formula Ta205 with density 8.74 g/cm3. The increase in the film thickness was
0
15.3 A/V. Thus, the oxide film obtained on the investigated sample had a thickness of 1530 ? 30 A (the
error is associated with inaccuracy of the establishment of the voltage and current density of oxidation).
Such agreement can be considered satisfactory, considering the possibility of error in the determination of
the energy losses of protons in tantalum oxide.
We should also note the promise of the use of the resonance of the reaction 018(p, a)N15 for the in-
vestigation of thick surface layers of metals. Figure 4 presents the spectra of a-particles from a sample
of zirconium oxidized in water vapor for 120 min. At Ep > 629 keV, a peak due to the resonance space of
a-particles distinctly appears. The successive increase in the energy of protons bombarding the target
leads to an increase in the depth at which the protons have an energy close to the resonance value. From
the spectra obtained in this way, according to the change in the contribution of the "resonance" a-particles,
we can establish the nature of the change in the oxygen concentration with the depth. The use of resonance
permits high resolution with respect to depth to be obtained, which is associated chiefly with the width of
the resonance and the nonmonoenergeticity of the protons in the beam. In our case the resolution was -320
and -270 A for zirconium and tantalum, respectively. The methods outlined permit an investigation of the
interaction of oxygen with various solids, in particular, with most metals and alloys used in reactor con-
struction.
The authors would like to thank G. K. Khomyakov, A. G. Strashinskii, and P. P. Matyash, for their aid
in the experiments.
1. A. Choudhury et al., Solid St. Communications, 3, 119 (1965).
2. G. Amsel and D. Samuel, J. Phys. Chem. Solids, 23, 1707 (1962).
3. C. Williamson, J. Boujot, and J. Picard, Tables of Range and Stopping Power of Chemical Elements
for Charged Particles of Energy 0.05 to 500 MeV, Rapport CEA-R 3042 (1966).
4. F. Ajzenberg-Selove and T. Lauritsen, Nucl. Phys., 11, 1 (1959).
5. G. Amsel et al., J. Phys. Chem. Solids, 30, 2117 (1969).
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Mathematical methods of treating experimental results are beginning to be widely employed in activa-
tion analysis [1, 2].
The possibilities of these methods are checked with the aid of synthetic mixtures, i.e., mixtures of
known qualitative and quantitative constituents. The preparation of synthetic mixtures of different concen-
trations often gives rise to technical difficulties. In the measurement of the radioactive decay character-
istics of a mixture in a natural experiment there are introduced, in addition to the error associated with
the probabilistic nature of the decay process and of the processes whereby the radiation interacts with the
detector material, additional errors associated with the operation of the specific electronic-physical re-
cording apparatus. Computer simulation of radioactive decay processes permits elimination of the addi-
tional errors and is less time consuming.
Using the terminology of [3], we consider in this article the computer simulation of the radioactive
decay of a mixture of isotopes.
Simulation of Radioactive Decay Curve
Quantitative Description of Process. At moments of time tk(k = 1, 2, . . . , m) separated by an in-
terval Atk the quantity of disintegrated nuclei Nk is recorded; the resulting numbers Nk constitute the radio-
active decay curve.
Mathematical Model of Process. We consider a mixture of radioactive isotopes (n isotopes in all)
which differ from one another in the parameter Xi, the decay constant. At the initial moment of time to the
number of radioactive nuclei of the i-th isotope (i = 1, 2, . . . , n) equals Noi. Each nucleus of any isotope
decays at some moment of time t (0 < t < on). The probability fi(t) that a particular nucleus of the i-th iso-
tope will decay during a time t, t + dt is given by the radioactive decay law:
fi (t)=die-~`t. (1)
1000 Algorithm. We determine the decay time of each
137
Gs
nucleus of the mixture by selecting random numbers from
a set of random numbers distributed according to law (1).
?~ .? a Mnsa We fix those cases when the nucleus decays in the interval
V. tk, tk + Otk. The resulting set of numbers Nk will con-
numbers with the above distribution. Several methods
25
Channel No.
random numbers with an arbitrary distribution law [4].
Since any computer can be employed to generate pseudo-
Fig. 1. Computer simulation of y-spectrum random numbers uniformly distributed in the interval [0,
of a three-component mixture. e) denotes 1] and satisfying the statistical criteria for randomness
experiment; and ^) simulation, and uniformity, the best method of obtaining random
Translated from Atomnaya Energiya, No. 5, pp. 459-460, May, 1971. Original letter submitted Jan-
uary 4, 1970; revision submitted July 23, 1970.
? 1971 Consultants Bureau, a division of Plenum Publishing -Corporation, 227 West 17th Street, New York,
N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without
permission of the publisher. A copy of this article is available from the publisher for $15.00.
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numbers with an arbitrary distribution law in our case is the method of change of variables (the inverse
function method). In this method a random number ~ from the set of random numbers uniformly distributed
in the interval [0, 1] is placed into correspondence with a random number r) from the set of random num-
bers distributed according to the law (1) by the formula:.
11= - 1 In (2)-
TT A detailed discussion and statistical analysis is given in [4] of the computer production of pseudo-
random numbers. By varying Noi and. the duration of the measurement Atk, we can obtain the decay curves
of a mixture of radioactive isotopes of arbitrary concentration with the requisite statistical accuracy.
Simulation of y-Spectrum of a Mixture of Radioactive Isotopes
Using a Spectrometer with a Multichannel Analyzer
Quantitative Description of Process. The y-quanta emitted by the isotope mixture are distributed
over the analyzer channels. The measurements result in a set of numbers Nk, the spectrum of the mixture
or the number of pulses in the k-th channel of the analyzer (k = 1, 2, -... , 1, where 1 is the number of chan-
nels).
Mathematical Model of the Process. We consider a mixture of radioactive isotopes emitting y-quanta.
The total number of radioactive isotopes equals n, and the activity of the i-th isotope we denote by Ai. The
probability that a y-quantum from the i-th isotope will be recorded in the k-th analyzer channel is com-
pletely determined by the distribution fi(k). The distribution function Fi(k) expresses the probability that
a y-quantum from the i-th isotope will be recorded in any of the analyzer channels from 1 to k:
k
Fi (k) /i (l),
j=1
Algorithm. The number of -y-quanta emitted by the i-th isotope in a time t equals
Ni =Ait.
From the inversion method the number k of the channel into which a y-quantum from the i-th isotope passes
is given by the inequality
By repeating the operation Ni times, we obtain the spectrum of the recorded y-quanta from the i-th isotope.
The y-spectrum of a multicomponent mixture is obtained by summing over all isotopes:
The distribution function is obtained from the so-called reference spectrum or standard. To this
end we record with good statistics the y-spectra of the individual isotopes and then normalize to the total
area of the spectrum:
/i (k)=Ni (k)/Y Ni (/).
j==1
Clearly, the possible errors arising upon simulation in accordance with the above formulas are de-
termined by the qualities of the particular pseudo-random number generator. For example, if the aperi-
n
odicity segment L is comparable with the quantity of random numbers. selected during simulation (L E Noi
n i=1
for simulation of a decay curve, or L ;zti E Ni for simulation of a y-spectrum), a systematic error will be
1_1
present in the results obtained.
The Minsk-22 computer was programed in accordance with the above algorithms. The figure shows
the simulated y-spectrum of a three-component mixture and the spectrum obtained experimentally using a
scintillation spectrometer with an AI-100 amplitude analyzer and a 70 x 70 crystal.
LITERATURE CITED
1. E. M. Lobanov, N. V. Zinov'ev, and A. G. Dutov, in: Activation Analaysis of Pure Materials [in Russian],
Fanlar, Tashkent (1968), p.26.
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2. B. G. Egiazarov et al., Program and Theses of Reports of 19th Annual Conference on Nuclear Spectro-
metry and the Structure of the Atomic Nucleus [in Russian], Nauka, Leningrad (1969), Part 1, p. 203.
3. N. P. Buslenko, Mathematical Simulation of Industrial Processes [in Russian], Nauka, Moscow (1964).
4. D.I. Golenko, Simulation aid Statistical Analysis of Pseudo-Random Numbers on Electronic Computers
[in Russian), Nauka, Moscow (1965).
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ESTIMATE OF EMERGENCY DOSES AT HIGH-POWER y-FACILITIES
E. D. Chistov, I. F. Spr.ygaev,
I. P. Korenkov, A. V. Terman,
and A. V. Sedov
Experience has shown that the operation of high-power y-facilities may lead to accidents or to emer-
gency situations. One way -in which emergencies arise is from failure of the interlock system.on the protec-
tive entrance door of the facility and the malfunction of the system signaling the condition of the radiator
and the dose rate levels [1, 2]. The serious consequences of such accidents make it necessary-to estimate
and predict'the external radiation dose received by a victim.
A typical high-power y-facility has a working chamber and . a. labyrinth. If.the interlocking and signal-
ing systems are defective the -,victim can open the.door to the labyrinth and approach the radiator along-the
path shown in Fig. 1.
t2
where Dl = P . dt tE Piti is the dose received during the time t2 - ti the victim is approaching the radia-
tor, D2 is the dose received during the time-t3 - t2 the victim'is in the vicinity of the radiator, and D3 = P
t
? dt -Piti is the dose received during the time t4 - t3.the victim is moving away from the working cham-
ber. It is necessary to know the dose. distribution along the path of the victim through the labyrinth in order
to determine these doses.
Fig. 1. Typical -plan of chamber in a high-power y-facility:
- ? - ? -)'axial line; : ) most probable path of victim; a, b,
c) levels of head, abdominal cavity, and bottoms of feet;
1) radiator; 2) .holder with source.
Translated from Atomnaya Energiya, No. 5, pp. 460-462, May, 1971. Original letter submitted May
11, 1970.
? 1971 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York,
N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without
permission of the publisher. A copy of this article is available from the publisher for $15.00. -
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V_ -
2
.~
0,2
7 6
9
10 1J
N
N N
16
If
1
1
19
Fig. 3. Distribution of -y-dose rate over ver-
tical cross sections of labyrinth from mea-
surements at 20 points. Cross sections (in
mm) as shown on Fig. 1: a) PT plane; b) GI
plane. Figures on curves are values of P',
in R/h. Activity of source 40,000 g-eq Ra.
6 181520 22
0 2 4 6 8 1012141
t, sec
as a
Fig.2. y-Dose rate Py
R from
function of path length
radiator through labyrinth.
8
7
3
2500
D9
00
11
9
4000
16
)
18
19
The dose at any point in the labyrinth can be found as the sum
of the doses from direct and back-scattered (reflected) radiation [3]
n
D 8.4At e-?:B hv, x, z)8.4A?t Si cos 8, di (?, E)
lab- R2 ( t Lx, R21+1
t=1
m where A is the y-equivalent of the radiator in mg-eq Ra; R is the dis-
tance from the radiator to the dose point (e.g., point K of Fig. 1) in cm;
the R1 ? are the distances from the radiator or the principal scat-
tering planes to the dose point in cm; x is the thickness of the concrete
shield in cm; B(hv, ?x, z) is the dose buildup factor; Si is the area of
a scattering surface in cm2; O is the angle between the normal to the
scattering surface and the direction of the incident radiation; t is the
duration of the exposure in hours, and a (O, E) is the albedo for y-radiation.'
The total dose received by the victim while he is approaching the radiator is given by the relation
r e u`. R (hy, ?x, z) Si cos ?idi (6, E) l t'_ t
1)f=0.831 8.4?A L f ( t)'
-t
R2 (2)t`I;i t-1 Ri-~1
i=1
where t2 - tl is the time in hours the victim spends in moving from the labyrinth entrance to the radiator,
and 0.88 is the conversion factor from roentgens to rads. The dose D3 is given by an expression similar
to (3) for the time in hours t4 - t3 spent by the victim in moving from the radiator to the labyrinth exit.
D-0.88D2+7.4 [(t2-t0+ (t4-t3)] C,
i-1
where C is the expression in square brackets in Eq. (3).
We have performed calculations and experiments to determine emergency doses fora facility having
an entrance labyrinth with three bends. The dose distribution in the labyrinth is shown in Fig. 2. Curve 1
characterizes the distribution at foot level (20 cm above the floor) and curve 2 at 1 m above the floor. In-
vestigations of the dose distribution in various vertical cross sections of the labyrinth show that after the
first bend the distribution is practically uniform, i.e., it is unaffected by the orientation of the radiator in
the working chamber (Fig. 3a, b).
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TABLE 1. Calculated and Measured Absorbed Doses. in Model of Human Body
Place of measure-
ment
Head 10.0
8.2
Gonads
37.0
23.0
Neck 16.2
16.0
Thigh
38.0
28.0
Chest 24.0
18:0
,Shin
27.0
20.0
Abdomen 29.0
20.0
To check the. calculations experiments were performe'dwith;a,phantom under conditions-simulating an
emergency situation at a radiation facility having :a Co60 radiator with-an activity of .4000 g-eq Ra. The
phantom was. a wooden model :of the human body, had :a density of,0.72,.g/c.m3, :and closely simulated soft
human tissue in composition. The :phantom had..a mass of 57 kg, was 170 cm high, and the sizes of its in-
dividual parts corresponded to the parameters of.the "standard",rnan. Thephantomhad22 holes 30 mm in
diameter filled.with wooden plugs having cylindrical cavities 12 mm long and 1.2 mm in radius. for alumi-
num phosphate dosimeters. The emission of the thermoluminescent dosimeters was measured in a special
heating device of the, DTM-2 type. The intensity of luminescence was recorded with a type FEU-33 photo-
multiplier. The calculated and experimental results are shown in.Table 1.
The appreciable divergences between calculation and experiment in many cases can be explained by
the shielding of certain parts of the phantom by structural members. Calculation showed that a dose of 1.2
rads is accumulated in moving through the labyrinth to the radiator and back. This quantity should be added
to the total dose.
Our calculational. procedure gives results which are adequate for practical purposes and can be used
to estimate radiation doses at high-power y-facilities during emergency situations.
LITERATURE CITED
1. E. D. Chistov et al., in: Scientific Papers of Institutes of.Work Safety VTsSPS [in Russian], No. 51,
Profizdat, Moscow (1968), p. 39.
2. L. L. Sokolina et al., in: Data from a Conference on Theoretical and Applied Radiation Safety (Novem-
ber 23-29, 1966) [in Russian], VTsNIIOT, Moscow (1968), p.242.
3. E. D. Chistov and A. V. Larichev, in: Scientific Papers of Institutes of Work Safety VTsSPS [in Rus-
sian], No.3 (29), Profizdat, Moscow (1964), p.49.
577
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USE OF SEMICONDUCTOR DETECTORS WITH A
p -n JUNCTION FOR THE DOSIMETRY OF X-RAYS
AND y-RADIATION IN THE LOW-ENERGY RANGE
V. K. Lyapidevskii and Yu. B. Mandel'tsvaig UDC 539.12.08
Diffusion-drift silicon or gallium arsenide semiconductor detectors with p-n junctions are widely
used for recording various types of radiation. The strong dependence of the sensitivity of these detectors
upon the energy of the radiation, particularly in the low-energy range (below 40 keV) has imposed limita-
tions on their use in dosimetry. Figure 1 shows the relative spectral characteristics of silicon detectors
having various thicknesses of the sensitive region; Fig. 2 shows similar characteristics for gallium arse-
nide detectors. The solid lines indicate the dependence of i/Pe upon the energy E of the radiation quanta,
where i denotes the signal at the detector output (this signal is proportional to the energy absorbed by the
sensitive region) and Pe denotes the dose rate received. The functions i/Pe were calculated with the for-
mula
ye 0u0-- /I mu
where h denotes the thickness of the sensitive layer; (?k)mB denotes the overall energy transfer coefficient
for air; and (pk)mZ denotes the overall energy-transfer coefficient for a material with the atomic number
Z.
The ?k values for GaAs were calculated with the data obtained with Cu29 [1, 21 and with the technique
described in [3]. The pk values for silicon were calculated with the data of [4, 5].
The position of the maximum depends strongly upon the thickness of the sensitive layer.* The form
of the curves undergoes only slight changes and the energy dependence of the sensitivity is very strong in
every case in the low-energy range. One must bear in mind that a compensation for the energy losses by
means of filters or semiconductor-luminophor combinations which give improved results in the range 40-
100 keV [6-9] is not very efficient in the low-energy range considered.
The dose of monoenergetic y-radiation or of a single-parameter x-ray radiation (i.e., a radiation
which depends upon a single parameter only, e.g., upon the attenuation coefficient ?) can be measured by
simultaneous determinations of two quantities [5]. We used two semiconductor detectors with p-n junctions,
namely a silicon detector and a gallium arsenide detector. The relative spectra characteristics i/Pe = f(E)
of these detectors are shown in Fig. 3.
The same characteristics are indicated in Figs. 1 and 2 by dashed lines. A comparison of the experi-
mental curves with the calculated curves leads to the conclusion that the thickness of the sensitive layer of
the silicon detector is close to 100 ?, whereas the thickness of the sensitive layer of the gallium arsenide
detector is several times smaller and amounts to 20-30 ?.
*The strong dependence of the position of the maximum of the i/Pe = f(E) curve upon the thickness of the
sensitive layer makes it possible to determine the layer thickness of a particular photodetector by com-
paring the experimental curve with a series of calculated curves. One can estimate the diffusion length L of
the charge carriers in the base material when h L (this occurs, e.g., in the Si used in solar batteries).
Translated from Atomnaya Energiya, No. 5, pp.462-464, May, 1971. Original letter submitted Feb-
ruary 20, 1970.
0 1971 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York,
N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without
permission of the publisher. A copy of this article is available from the publisher for $15.00.
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01101,75283540 57 71 82 93: 107 124 ?. (keV) D 10715283540 57 71 82 93 107 124 E, (keV)
Fig. 1 Fig. 2
Fig. 1.. Relative spectral characteristics of. diffusion-drift silicon detectors with various
thicknesses of the sensitive layer: 1, 214, 5) calculated. curves: for sensitive layer thick-
nesses of 0.01, 0.1, 1, and 10 mm, respectively; 3) experimental curve.
Fig. 2. Relative spectral characteristics, of diffusion-drift gallium arsenide detectors: 1, 2,
4, 5) calculated curves, for sensitive layer thicknesses of, 0.01, 0.03, 0.1, and 1 mm, respec-
tively; 3) experimental curve.
13
37 28 18
22
40 35 2,5
2,5 tZ
lq
Fig. 3 Fig. 4
Fig. 3. Relative spectral characteristics of silicon and gallium
arsenide detectors at a radiation energy Eeff. < 40 keV.
Fig. 4. Diagram for the determination of the dose rate from simul-
taneously measured currents of two detectors, a silicon detector
(i2) and a gallium arsenide detector (i1).
In accordance with [5], we indicate the ratio Pe/il on the vertical axis of a rectangular coordinate
system and the ratio i2/il on the horizontal axis, where i2 and it denote the output current of the Si and GaAs
detectors, respectively. Figure 4 shows the relations obtained. (The detectors were operated in a short
circuit at t = 20?C.) In order to measure the exposure dose rate, the two detectors were placed in a beam
of the radiation to be measured and Pe/ii was determined from the measured i2/il value as shown by the
arrows in Fig. 4. The Pe/ii value obtained was multiplied with the current i1 measured' in this experiment
and thus, the exposure dose rate Pe could be determined. The sensitivity of the detectors to x-ray radia-
tion with Eeff = 100 keV amounted to (0.7-1.3) ? 10-9 A/R/min ? cm2. The relation i = f (Pe) is linear in the
range 0.5-400 R, under these conditions.
The method of simultaneously measuring two quantities can also be used for the dosimetry of a radia-
tion with an unknown spectrum. However, in this case (contrary to the previously considered case) errors
can be made in the dose determination and the magnitude of the errors depends upon the radiation spectrum.
The greatest possible error in the Pe determination can be graphically estimated with Fig.4 [5]. In the en-
ergy range considered, an error is made when the radiation spectrum consists of two lines with quantum
energies of 40 and 10 keV. The relative error S = (Pmax - Pe)/Pe in the Pe determination is less than 35%
in this case.
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It can be shown that the maximum error which is made in exposure dose measurements of a homoge-
neous radiation with the aid of the setup of [10] (the, quantity alil f a2i2 is measured at the output of this
setup) is greater than or equal to the maximum error 6 determined from Fig. 4.
When. the above method is used, the error is usually much smaller than that determined from Fig. 4.
The error vanishes in the case of single-parameter spectra. One can therefore hope that the above-de-
scribed method of simultaneous measurements of two quantities will be widely used in dosimetry.
1. H. Jones, Physics of Radiology [Russian translation], Atomizdat, Moscow (1965).
2. G. V. Gorshkov, Penetrating Radiation of Radioactive Sources [in Russian], Nauka, Leningrad (1967),.
3. A. M. Gurvich et al., Transactions of the All-Union Scientific Research Institute for Medical Instru-
ments and Equipment [in Russian], No. 5, 40 (1962).
4. Yu. K. Akimov et al., Semiconductor Detectors for Nuclear Particles and Application of these Detec-
tors [in Russian], Atomizdat, Moscow (1967).
5. N. G. Volkov and V. K. Lyapidevskii, Pribory i Tekh. Eksperim., No. 5, 86 (1968).
6. L. Hollander, Rev. Sci.Instrum., 28, 322 (1957).
7. N. Bailey and G. Kramer, Radiation Res., 22, 53 (1964).
8. A. N. Krongauz et al., Meditsinskaya Radiologiya, No. 9, 78 (1970).
9. Yu. B. Mandel'tsvaig and V. G. Epishev, Novosti Meditsinsko Tekhniki, No. 2, 123 (1966).,
10. S. P. Vershinina, A. Ya. Berlovskii, and Yu. A. Tsirlin, Atomnaya Energiya, 24, 262 (1968).
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PROTON CURRENT FOR EQUILIBRIUM PHASE DECREASE
ALONG A LINEAR ACCELERATOR
A. D. Vlasov UDC 621.384.64.01
Calculations of the proton current in a linear accelerator for stable and neutral equilibrium of the
?
accelerated bunches showed that the;current increases from hundreds of milliamperes to about 10 A [1, 21
when the equilibrium phase is increased from 35-45? to 85-90?. However, an increase in equilibrium phase
requires an increase in the accelerating field and. in the length of the accelerator, and can be tolerated only
over a limited section. This paper discusses a linear accelerator with an equilibrium phase which is initial-
ly 85-90? and which gradually decreases along the accelerator.
The changes in particle energy, equilibrium phase, and other parameters along the accelerator are
assumed to be so slow that the bunches at each energy can be considered in a stationary approximation.
The state of a bunch at various energies is tied in with stability and charge conservation conditions and
also with the corresponding relation for the variation of the phase area of the bunch.
Only beams and bunches of constant density are discussed in the following. Of course, the velocity
spread in an injected beam must correspond rather well to the distribution.of velocities in the bunches dis-
cussed in order to avoid marked loss of particles.
(p cos (pso-sin
W
(Do ((P) = sin Wso
where co is the phase of the accelerating wave; cpso = arccos (dW/eEmdz) is the nominal equilibrium phase;
e and W are the particle charge and energy; Em is the amplitude of the accelerating wave; and z is the
longitudinal coordinate.
Using the well-known disc model of a bunch, we shall represent it in the form of a cylinder of fixed
radius r with a charge density p (in the laboratory system) which depends only on 9. Then the potential of
the intrinsic charge of the bunches, which follow one another with a spacing Oz =)9X, is expressed by [2]
vi I W-m' IVY
U(q)=aG [e- a -{-e a 2?e a P(p')dq'.
(Pf e a - 1
Here, cpi and 9f are the boundaries of the bunch; a = 27nfl - (32/kaA is the reduced radius of the bunch,
G = /41reoEmsincpso; A is the wave length; R = v/c; v is the particle velocity; c is the velocity of light; and
k is a factor of the order of one which depends on the ratio of the bunch and aperture radii to the quantity
f3A. We set coif = cpc ? ae.
Setting p(co) = po = const and performing the integration, we express the total potential of the acceler-
ating and bunch fields as
_ sh --e
(W)-(Do+U-~co S(10- in (p - 2a2Gpo a ch W T+const.
sh
a
We write the condition for the conservation of the current I and of the bunch charge Q = IA/c:
Translated from Atomnaya Energiya, No. 5, pp. 464-466, May, 1971. Original letter submitted May
25, 1970.
C 1971 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York,
N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without
permission of the publisher. A copy of this article is available from the publisher for $15.00.
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Asa
2
of=1 0,3
0,1
0.
3
i
7
-
1 2 3 4 5 6 7 p/pi 8
Fig. 1. Lower boundaries of allowed regions of vari-
ation" of equilibrium phase and of bunch phase length
for various cases.
BX IA5 poir2.2ae -n = - =const. (2)
We substitute in Eq. (1), expressions for
G and p0 based on condition (2):
(D ((P) = W cos (P80- sin p- 60a7?1
sin (pso Emr2 sin Aso
n 1
sh a e
X n/ ch a +const.
esh-a
For stability of phase oscillations, it is
necessary that c' ((of) 0 at the trailing edge of
a bunch, or, considering Eq. (3),
t \
11
6051 sh (a -e she
COS cpso-cos (Pc-ae)+ ? ? G0. (4)
Emr2 n e
sh -
a
The bunch boundaries are also related through the equation 4 (co f) = 1) ((pi). Hence,
ry~ sin as
cos 'YSO= as COS q1C
The area occupied by the image of the bunch on the (pW phase plane is expressed in the
(cpi -cpf) x 2(W-Ws)c. But
dW SX ? mock 02Q
W -Ws- ds 2nc (P _ - n 2c- (1- 2)3/2 v(Df -(D ((P)
where Slp = 27reEm(1 -/32)3/2sin,cps0/m0X/3; m0 is the rest mass of the particles; and 4)f = cI((pf) =[(1;((pf)
+ 4)((pi)1/2. Thus
V 2.8ae VeEmX sin Aso i (D -(D (Wc)
=
nmOC4 P 3/2 f
Here, and subsequently, p = m0c(3f1 - /32 is the particle momentum and pi = p/pi.
As is well known, the phase area in the (pW plane is an invariant, but the effectively filled phase area
gradually increases because of random small errors in the accelerator with V - 02(pl). Calculating -If
- 4, ((pc) by means of Eq. (3), we obtain
1
it
V2 a2e2Emps 60X1 ash a -e/ t11e-1
T4 .?. ,,4 [(1 -cos ae) sin (pc - E r2 ti =const.
sh
a
Let the variation of the parameters along the length of the accelerator be
=f (Pt)
y =+I2(P(), ri =X(P1), Emi =F(P(), Ti
POF -2 (1-cosae) sin (pc- I1f-. rz n (che-1)=
;h-
a
nrl Emi
It 60X
Considering (5), the inequality (4) and Eq. (6) then take the form
I nai1hesh (a -e)
S XfFPi,
w. sh a sin as C sin gc - (1 - ctg as / cos Tc]
l n ll
a2e2, lai as sh (a -el 2Cjp4
(10)
Inequality (9) and Eq. (10) determine the achievable current I (or I/I1) and the constant C through the
initial values of a, ae, (pc.
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For fixed I/Ii, the inequality (9) and Eqs. (5) and (10) determine the region of allowed values of pi,
4 so, which is bounded from below by the curve cpso = ,~pso(Pi) defined by Eqs. (5), (9), and (10). The allowed.
pi, ac and pi, (pc regions are thereby determined also. The limiting curves are calculated more easily by
assigning values to ae and finding cpc and pi from the equation system (9), (10) and then cpso from (5).
Ordinarily e > 3 and (ir/a) - e ? 1 over the greater part of an accelerator so that
she h(n che-fshe) :t; 1 (11')
ShA ? a / Shy a -
a a
In the calculation of typical limiting curves below, 'it .is -assumed that
'~=Pil4 /3., . :.I o=.2/3x=P1 f=F=i P1 ) , - (12)
When Eqs. (11) and (12) are valid, the calculations are significantly simplified.
We consider the injection of a continuous, uniform beam. In such a beam there is no longitudinal re-
pulsion, and the achievable current is determined only by the subsequent separation 'into, bunches. To re-
duce particle loss to a minimum, it is necessary to set cpsoi = 90?. Then 2 (ac)- = 360?, cpci = 90?, and we
find I = Ii, C = 1 from Eqs.. (9) and (10). Figure 1 shows the limiting curves for the variation of the equilib-
rium phase (Pso and of the bunch phase length-(pi - cpf = tae along an accelerator as a function of pi, calcu-
lated for ai equal to 0.1, 0..3, and 1.0 (curves 1). Note that ai = 0.32 for our 1-2 and I-100 accelerators since
ri = 4.75 mm, 1i = 0.0387, A = 2.02 m., and k = 1.2. Note also that when Wi = 0.7 MeV, the value p/pi = 8
corresponds to W = 46 MeV.
For the parameters of the I-100 accelerator (in which Em = 1.58 MV/m), Ii = 0.92 A. This current is
low in comparison with the current for neutral initial bunch equilibrium and (pso = 90? [2]:
02Em%k2 (i +a2)
IR \ 2 ) 240a
which is 10.1 A for the I-100 accelerator. Generally, when /32 1, i.e., for comparatively large reduced beam radii.
Also shown in Fig. 1 are the curves 1', which were calculated for ai = 0.3 and the same regime as for
curves 1 but with l - 1. It is clear that inaccuracies in the accelerator reduce the allowable regions.
The relative smallness of the current given by Eq. (13) for injection of a continuous beam results from
the comparatively large resultant bunch length, which is close to the length of the separatrix. As is well
known, the maximum current is achieved with shorter lengths [1] in the case of bunches of constant density.
Equations (5) and (9) make it possible to construct the curve I= I (di Ei) and to find the maximum I. For exam-
ple, let ai = 0.3. Then for values cpsoi =90 and 85?, we find (I/Ii) max = 3.86 and 3.16 respectively for (a ei) = 117 and
106 ?. 1111=0.92 A, I = 3.5 and 2.9 A. We find C = 0.251 and 0.176 from Eq. (10), and plot the limiting curves 2 and 3.
In designing an accelerator with some given variation of equilibrium phase (or of associated para-
meters), the given variation replaces Eq. (9) and makes up an initial equation system together with Eqs. (5)
and (10). The validity of any parameter variation is easy to evaluate by comparing it with the limiting
curves. Thus the constant bunch length ((Pi - cpf)/3A/2ir proposed in [3] is unrealizable for Eqs. (12) because
the curve ae)3 = const (curve 1") lies below curve 1. At the same time, the validity of constant equilibrium
phase cpso = cpsoi = const and of constant bunch phase length (pi - cpf = 2(ae)i = const is immediately apparent.
It is necessary to consider the parameters calculated above for cpsoi = 90? only as theoretical limits.
In practice, we always have (pso < 90?
LITERATURE CITED
1.
A. D. Vlasov, At. Energ., 27, 238
(1969).
2.
A. D. Vlasov, At. Energ., 29, 141
(1970).
3.
I. M. Kapchinskii and V. A.. Teplyakov, Pribory i Tekh. Eksperim., No. 4, 17
(1970).
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ABOVE E0 IN AN ISOCHRONOUS CYCLOTRON
L. A. Sarkisyan UDC 621.384.633
Nonlinear effects resulting from the periodicity N of the magnetic field play an important role in iso-
chronous cyclotrons with spatial variation of the magnetic field. Resonance relations between the free
betatron oscillation frequencies Qr and QZ and the operating focussing harmonic of the field periodicity N
lead to the appearance of ideal nonlinear resonances of the form
PQr ? 4Qz = N, (1)
where p, q = 0, 1, 2. . . ; I p + qJ determines the order of the resonance.
Radial nonlinear resonances (q = 0) are the most dangerous because they are of lower order. As is
well known, the frequency Qr of radial betatron oscillations increases in isochronous cyclotrons during ac-
celeration and passes through a series of resonance values Qr = N/p. The lowest value Qr = 1 occurs at
the center of an accelerator with a solid structure (in ring cyclotrons, Qr is somewhat greater than 1 and is
determined by the injection energy), and the maximum value of Qr is determined by the resonance order p.
Depending upon the value of p, these resonances (Qr = N/p) may lead to a considerable increase in the am-
plitude of the radial oscillations of the particles. Thus the particle kinetic energy which can be achieved in
an isochronous cyclotron is determined by the resonance values of the radial betatron oscillation frequency
and, from linear theory, is approximately
Ti =E0 Qr 1 ,
` -
3 sr 2
7
1+ _1_V2
where E0 is the rest energy of the particles; s is the variability of the fundamental harmonic of the field;
jc is a parameter of the Archimedes spiral; and r is the instantaneous radius of the orbit.
Calculations carried out in [1] showed that for a free radial oscillation amplitude of a few centimeters
and the acquisition by the particles of an energy of -300 keV per turn, the ideal nonlinear resonance of the
fourth order (p = 4) leads to practically complete loss of beam, and the maximum energy of protons in iso-
chronous cyclotrons is restricted by the nonlinear resonance Qr = 2. Note that in the electron analog II at
Oak Ridge, the resonance Qr = 2 occurred for particles with very small radial oscillation amplitudes (for
protons, an amplitude of the order of a few millimeters) [2].
Passage through the resonance Qr = 2 and further acceleration brings with it considerable reduction
in beam intensity and the need to overcome difficulties in shaping the magnetic field because the nonlinearity
of the average field and the variability increases with radius. In addition, the beam extraction system be-
comes considerably more complex and its efficiency falls significantly. Therefore, in isochronous cyclo-
trons designed for maximum energy, the radial betatron oscillation frequency is selected to be Qr = N/q
= 8/4 = 2, and the maximum proton energy in the accelerator is somewhat less than the rest energy E0 de-
pending upon the values of the parameters e, N, and ~t.
Thus it is impossible to accelerate protons in an isochronous cyclotron to a kinetic energy above -E0,
i.e., for Qr > 2. It is therefore necessary to use a cascade method of acceleration where each isochronous
cyclotron is the injector for the next one. We shall call such an accelerator a cascade isochronous cyclo-
tron.
Translated from Atomnaya Energiya, No. 5, pp.466-468, May, 1971. Original letter submitted May
11, 1970.
? 1971 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York,
N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without
permission of the publisher. A copy of this article is available from the publisher for $15.00.
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TABLE 1. Cascade Cyclotron Parameters
Cascade ac-
celerator
Ho,oe
w cm
Q,
Qr
cm
ri, cm
rf, cm
ei
W,MeV
fo,MHz
First
6500
8
7
0,2
1 < Qr G 2
447
0
385
0,245
823
9,88
Second
3000
12
7
0,2
- 2 G Q? < 3
1042
902
982
0,157
1670
4,56
Third
1500
16
7
0,2
- 3 G Qr G 4
2084,
1965
2018
0,105
2523
2,28
Note: Ho is the magnetic field intensity at the accelerator center for r= 0; r,, = Eo/eHo; ri and rf are the initial
and final acceleration radii; fo is the rotational frequency of the particles.
Note that the concept of cascade acceleration has recently found extensive application for increasing
particle energies in accelerators of a different type. In the sub-GeV region of cyclotron acceleration, there
is the well-known Zurich design for a ring cyclotron producing,'-600 MeV protons in which it is proposed
to use a cascade of two accelerators. The first is an approximately 70 MeV, four-sector, "Thomas" cyclo-
tron operating in the radial betatron oscillation frequency range 1.0 Qr 1.1, and the second is an eight-
sector ring cyclotron operating in the frequency range 1.1 2, is achieved by the selection of a definite frequency range for the radial betatron
oscillations Qr in each accelerator. The frequency range of the radial betatron oscillations in the first,
second, third, etc., isochronous cyclotrons of the cascade is selected to be respectively -1 Qr N/p = 8
/4 = 2, -2 < Qr : N/p = 12/4 = 3, ^-3 < Qr N/p = 16/4 = 4, etc., where the upper value of the frequency
Qr in each cyclotron corresponds to an ideal nonlinear resonance of the fourth order (p = N/Qr = 4), which
is determined by the appropriate selection of the periodicity N of the magnetic field.
The first isochronous cyclotron may be either solid or of the ring type (with external injection of the
particles); all subsequent cyclotrons are of the ring type.
Transfer of the beam from one, accelerator to another is facilitated by the fact that the resonance
coupling between radial and azimuthal motion arising through the nonlinear resonance of the fourth order for
Qr = 2, 3, 4, etc., will be used inthese cyclotrons to achieve high-efficiency beam extraction. Note that with
careful optimization of beam characteristics and extraction system, a beam extraction factor of about 95%
was achieved with the help of the nonlinear resonance Qr = 2 at the electron analog II at Oak Ridge [4].
The assurance of a large radial increase per turn of the order of 1 cm at the internal radii makes it
possible to use an electrostatic or magnetic channel with a central plate for transferring the beam during
injection into the ring cyclotrons.
Table 1 gives possible parameters for an accelerator cascade which would make it possible to ac-
complish cyclotron acceleration of protons to a kinetic energy -2.5 GeV.
Further increase in proton energy by means of the version of a cascade isochronous cyclotron just
discussed will be determined to a considerable extent by economic considerations because each cyclotron
provides an increase in proton energy by approximately a constant quantity equal to E0. In this regard, an-
other version of the cascade isochronous cyclotron is feasible; this is one in which the energy increase in
the second cyclotron is not roughly the rest energy of the proton but double or triple it. This can be ac-
complished if the change in radial betatron oscillation frequency in a cyclotron is not one but two or three.
With an increase in the axial rigidity of an isochronous cyclotron (Qz >_ 1), the average beam current
increases by about two orders of magnitude [5, 6].
It is well known that the threshold for the creation of heavy particles by protons on nuclei is, 1.1 GeV
for K+ and A?, 1.3 GeV for E+ and E?, and 1.8 GeV for K- and K0. With an energy of -2.7 GeV, therefore,
the accelerator just discussed becomes a high-intensity K-meson "factory" which is competitive with linear
accelerators.
Note that existing and rebuilt synchrocyclotrons with energies of 600-1000 MeV and 700-1000 MeV
linear accelerators under construction can be used.as a first stage of the cascade along with an isochronous
.cyclotron.
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 Declassified in Part - Sanitized Copy Approved for Release 2013/02/25: CIA-RDP10-02196R000300080004-0
1. V. P. Dmitrievskii, V. V. Kol'ga, and N. I. Polumordvinova, in: Proceedings of the International Ac-
celerator Conference, Dubna, 1963 [in Russian], Atomizdat, Moscow (1964), p. 833.
2. J. Martin and J. Mann, Nucl. Intr. and Methods, 18, 19, 451 (1962).
3. H. Willax, Intern. Cyclotron Conference, Oxford (1969).
4. J. Martin et al., Proc. of the Intern. Conf. on Sector-Focussed Cyclotrons and Meson Factories, CERN
(1963), p. 52.
5. V. N. Anosov et al., At. Energ., 25, 537 (1968).
6. M. Gordon, Nucl. Instr. and Methods, 58, 245 (1968).
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TURBULENT HEATING AND CONFINEMENT OF ?A PLASMA
IN A TOROIDAL TRAP OF MULTIPOLE TYPE
B. A. Demidov and S. D. Fanchenko _UDC 533.9.16:621:039.634
As is known, turbulent heating,by a current. differs, from the'usual Joule heating in that the former
uses strong electric fields.for which the plasma resistivity is anomalously large-because of the develop-
ment of very-small=scale.instabilities. To'-facilitate passage. of the turbulent-heating current along the
circuit of the torus in the "Vikh'r'-3" device (Fig. 1), in-addition to the field B(P, a longitudinal field Bz