SOVIET ATOMIC ENERGY VOLUME 18, NUMBER 6

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• Document Page Count: 152
• Document Creation Date: January 4, 2017
• Document Release Date: September 24, 2013
• Sequence Number: 6
• Publication Date: June 1, 1965
• Content Type: REPORT

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? /
Voilime 18, Number 6
'June, 1965-
7."
ATOMHAR 3HEPrIIR
(ATOMNAYA siNERGIYA)
TRANSLATED FROM 'RUSSIAN
CONSULTANTS 'BUREAU
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ATOMNAYA ENERGIYA
A
EDITORIAL BOARD. I. Alikhanov
M. G. Meshcheryakov SOVIET ATOMIC
A. A. Bochvar M. D. Millionshchikov
I. N. Golovin
V. B. Shevchenko ENERGY
N. A. Dollezhal' (Editor-in-Chief)
V. S. Fursov P. N. Palei
V. F. Kalinin D. L. Simonenko
N. A. Kolokol'tsov V. I. Smirnov
(Assistant Editor) A. P. Vinogradov A translation of ATOMNAYA ENERGIYA,
A. K. Krasin N. A. Vlasov
A. I. Leipunskii (Assistant Editor) a publication of the Academy of Sciences of the USSR
V. V. Matveev
? 1966 CONSULTANTS BUREAU, A DIVISION OF PLENUM PUBLISHING
CORPORATION, 227 west 17th Street, New York, N. Y. 10011
Volume 18, Number 6 June, 1.965
CONTENTS
Phase Grouping of a Beam of Charged Particles During Capture into Acceleration
in the OIYaI Synchrophasotron?G. S. Kazanskii, A. I. Mikhailov, N. B. Rubin,
RUSS.
PAGE PAGE
and A. P. Tsarenkov
709
555
Improvements in The Cern Synchrocyclotron Over the Past Three Years?P. Lapostoll
714
559
Reduction of y-Ray Background from Induced Activity at CERN Proton Synchrotron
by Using Low Activity Absorbers?M. Barbier
720
565
Flux of Secondary Neutrons Produced by 660 MeV Protons in Shielding
?V. A. Kon'shin, E. S. Matusevich, S. S. Prokhorov
728
573
The Effect of Delayed Neutrons on the Time of Establishing a Stable Fission Chain
?V. F. Kolesov
734
578
The Shape of the Spectrum of Moderated Neutrons in Absorbing Media?V. N. Avaev.
741
584
A New Method of Reconstructing True Spectra?A. N. Tikhonov, V. Ya. Arsenin,
A. N. Dumova, L. V. Mayorov and V. I. Mostovoi
747
588
Measurement of Neutron Spectra in Nickel, Iron, and Stainless Steel?I. I. Bondarenko,
V. G. Liforov, V. N. Morozov, M. N. Nikolaev, V. A. Parfenov, and V. A. Semenov
752
593
Self-Diffusion in the a and 13 Phases of Uranium?A. A. Bochvar, V. G. Kuznetsova,
V. S. Sergeev, and F. P. Butra
761
601
Study of the Mechanical Properties of Beryllium?N. N. Davidenkov, B. A. Sidorov,
L. M. Shestopalov, N. F. Mironov, N. M. Bogograd, L. A. Izhvanov and S. B. Kostogarov.
768
608
Thermographic Investigation of UO2, UC1, and KCI Ternary and Binary Systems
?N. S. Martynova, I. V. Vasil'kova, M. P. Susarev
777
616
Removal of Radioactive Isotopes from Sewage?F. V. Rauzen and Z. Ya. Solov'eva
784
623
NOTES ON ARTICLES SUBMITTED
Waveguide Accelerator-Buncher Intended to Produce a Monokinetic Electron Beam
?G. I. Zhileiko and V. A. Snedkov
789
627
Maximum Efficiency and Limiting Current of an Electron Beam in a Heavy-Current
Waveguide Accelerator?G. I. Zhileiko
790
628
Dose Rate from a Unidirectional Source of Gamma Quanta Close to the Ground-Air
Interface.?Yu. I. Bublik, S. M. Ermakov, B. A. Efimenko, V. G. Zolotukhin and t. E. Petrov.
791
629
Ages of Neutrons from Mono-Energetic and Multi-Energetic Sources in a Uniform Moderator
?D. A. Kozhevnikov
793
630
Annual Subscription: $95
Single Issue: $30
Single Article: $15
All rights reserved. No article contained herein may be reproduced for any purpose whatsoever
without permission of the publisher. Permission may be obtained from Consultants Bureau, A
Division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011, U.S.A.
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CONTENTS (continued)
RUSS.
PAGE PAGE
HealTranSfer and Temperature Fields in Bundles of Rod-Shaped Heat-Emitting Elements,
Parallel to the Laminar Flow of a Liquid in which They Are Immersed
?M. Kh. Ibragirnov and A. V. Zhukov 794 630
The Role of Diffusion in the Migration of Radioactive Contaminants?V. M. Prokhorov 796. 631
LETTERS TO THE EDITOR
A 10 MeV WavegUide Synchrotron?A, A. Vorob'eV, A. N. Didenko, A. I. Lisitsyn,
B. N. MorOz0v, Yu. I. POtekhin, L. G. Salivon, and R. M. Vilatova 797 633
'Passage Through the Critical Energy in an Automatically Controlled Accelerator
?g, A. Zhil'Icov -799 634
Particle 1.,o8Ses Due to Passage Through Nonlinear Resonances in Accelerators and Storage
Devie es ?A A. K olornenskii 803 636
Effective Method of Solving the Two-Dimensional Diffusion Equation for Square
and Hexagonal Cells?G. I. Marchuk and V. P. Kochergin 806 638
Critical Thermal Load in Bunches With Spacer Grids?A. S. Korekov and Yu. D. Barulin 810 640
Thermoelectric Properties of Polycrystalline Uranium?A, A. Tsvetaev,
YU. N. Golovanov, R. K. Chuzhko, and I. V. KirillOv ? ? 813 642
Effect of Crystallization Rate and Annealing on the Plastic Properties of High-Boron Steel
R. V. Grebennikbv and A. V. Chirkin 816 .644
Precipitation Conditions for Uranyl Ammonium Arsenate and Some of its Properties
?M. A. Korenev, B. V. Nevtkii, Z. P. Zorina, Ts. L. Ambartsurnyari, and N. G. Nazarenko. 819 647
Some Possible Methods for Preparing Radioactive Isotopes?N.13, Rudenko, and A, M. Sevaseyanov 822 649
Optimum Specific y-Activity of a Quasi-Homogeneous Radiation Chemical Plant
?V. A. gletekOV 825 650
N-I-P Germanium Detector Features High Resolution for Low-Energy arid Mediurn-Energy Gammas
?L. V. MasloVa, 0. A. Matveev, S. M. Ryvkin, I. A. Sondaevskaya, N. B. Strokan 829 654
Radiation Hazard in Irradiated Metal Specirhens?M, M. Krasnoshchekov 832 656
Tracer Elements Aid in Evaluating Radioactive Anomalies in Arid Regions?V. M. Konstaritinov. . 834 657
Contamination Of Flora by Radioactive Fallout?O. G. MironoV . . 837 659
SCIENCE AND ENGINEERING NEWS
Nuclear Electric Power Stations in itaiy?v:v. S trekol 'nikoV , A.N. Grigoryants, Pancheriko. 840 662
A Trip to England?O. G. Kazathkovskii 844 664
Standardization of Nucleonic Instrumentation?I. AuZout 847 666
Participation of the USSR in the CEI TeChnical'Comnaittee 45,?N, M. kitaeV and V. V. Matveev. 851 669
The Russian press date (podpisano k pechati) of this issue was 6/14/1965.
Publication therefore did not occur prior to this date, but must be assutned
to have taken place reasonably soon thereafter.
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PHASE GROUPING OF A BEAM OF CHARGED PARTICLES
DURING CAPTURE INTO ACCELERATION IN THE OIYaI SYNCHROPHASOTRON*
(UDC 621.384.611)
G. S. Ka.zanskii, A. I. Mikhailov, N. B. Rubin, and A. P. Tsarenkov
Translated from Atomnaya tnergiya, Vol. 18, No. 6,
pp. 555-559, June, 1965
Original article submitted June 24, 1964
The authors give a method for improving capture by means of preliminary imposition of a high-frequency ac-
celerating field. The frequency of this field is varied according to the law of orbital deflection in "quasi-betatron"
conditions. The accuracy and possible errors in the establishment of these conditions are estimated, and experi-
mental results given.
It is known that a beam of particles injected into the chamber of a synchrophasotron has a certain angular and
energy scatter.t This leads to the formation of a fairly complex structure in the proton beam in "quasi-betatron"
conditions, causing some difficulty in effecting capture of the particles into synchrophasotron acceleration. It is
known that 17% of the injected beam is captured from quasi-betatron into synchrophasotron conditions. This capture
efficiency is determined [1] mainly by two factors:
1) particles with large amplitudes of radial betatron oscillation are unavoidably lost during capture and forma-
tion of a beam of accelerated particles in synchrophasotron conditions;
2) in synchrophasotron conditions, the particles are captured from part of the annular bunch, limited by the
phase dimensions of the region of stability. For the OIYaI synchrophasotron, the latter extends about 2000
.
In this paper we discuss a method for improving the efficiency of capture into synchrophasotron conditions by
grouping the particles in the azimuthal direction [2]. This grouping is effected by bunching together that fraction of
the particles which, in normal conditions, find themselves outside the stability region. If we create a stability region
of which the phase width is 3600, and then displace it inwards along the radius and draw this region together to normal
azimuthal dimensions, we can, in certain conditions, improve the efficiency of capture into synchrophasotron con-
ditions. As far as we know, a similar program was studied for the Bevatron (USA).*
Let us investigate the practicability of realizing this program. The equilibrium motion of the particles in ac-
celerators of the "race-track" type is given by the following equations [3]:
dEs nV 21( OF
ecp COS (s dt 11, Ils?c at '
dEs Es ( w Bo ;
1_
dt Ks \, o
s =-- asll s Esp2
zs = 0,
no
* OIYaI = United Institute of Nuclear Research.
t For the OIYaI synchrophasotron, the energy scatter A W/W = 0.5%; the angular scatter a = ?5'.
* The calculations of the Bevatron group were not made available to the present authors.
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(1)
709
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Fig. 1. Law of variation of frequency
of accelerating potential in phase-
grouping regime.
Fig. 2. Law of variation of frequency of
accelerating potential during initial part
of accelerating cycle with phase-grouping.
AB) Law of frequency variation for phase-
grouping conditions; BC) law of frequency
variation for synchrophasotron conditions.
where Es = equilibrium energy, Bo = magnetic induction for z = 0 and some fixed radius in the electromagnet sectors,
Vo = amplitude of accelerating potential, cps = equilibrium phase, Fs = flux through area of orbit, c = velocity of
light;
K s = 1 + ast1 =
4L
(where n = index of fall in magnetic field, I = 1 + ),
rs
IIs = 271-rs + 4L = perimeter of equilibrium orbit (where rs = equilibrium radius, L = length of straight-line section);
= frequency of accelerating field.
We require that during capture the equilibrium energy shall not vary, i.e., we shall assume that Es = o. We
shall neglect the effect of the vortical electric field 3Fs/at.
From these conditions it follows that cos cos = 0 and r, since iti the con-
trary ease it can be cansidere&Sirnply aS a 6 -function,
and it can be assumed that ui z(ti).
Processing of the two experimental spectra, cor-
responding to one and the Same true spectrum, meas-
ured by means of selectors having different resolving
powers. The neutron pulses were recorded by an anal-
yzer with a channel width of r 6 bisec.
The first experimental ("apparatus") spectrum is obtained on a selector with quite good resolution a = al =
= 4 1.1sec (s = 0.1 cm; R = 6.76 cm: co = 0.104 ? 10-2). Since in this case 0< r , the "true" spectrum z(1)(t) should
differ slightly from the measured spectrum. The function ui(1) is plotted in Fig. 1 as the Solid line.
The second experimental spectrum ui(2) is obtained on a selector with a worse resolution a = 02 = 17 psec
(s = 0.1 cm; R = 6.76 cm; co = 0.224 ? 10-3 psec). The function ui(2) is plotted in Fig. 1 by the dash-dot line and,
obviously it is a smoother curve than ui(l).
In order to verify the validity of the assumptions about the kernel of the integral equation and the accuracy of
the experimental data about the spectrum 14(2), the integrals
2 3 4 5 5 6 9
Ratio of neutron energy to temp. of medium
Fig. 4. Directional ( ) and scalar C-----)
neutron fluxes in a uranium-water lattice:
(P?ti)2
2cq u(I) (1,) dl'
? pr2.77J1
0)
were evaluated.
If ui(1), in fact, can be assumed to be the true spectrum, then the function ui(3) should coincide with the meas-
ured spectrum ui(2) . The function ui(3) is plotted as a dotted curve in Fig. 1. It is very close to the experimental
spectrum ui(2), which indicates the validity of the specification of the kernal, but somewhat smoother than ui(2).
The discrepancy, which is observed particularly clearly for channels with numbers i30-35, can be explained by the
measurement errors. Consequently, it is not Very hopeful that by solving the integral Eq. (2) the true spectrum ui(1)
will be successfully and accurately reconstructed by the spectrum ui(2) in the vicinity of
In order to demonstrate the feasibility of the method of regularization of the solution of the integral Eq. (2),
the spectrum ui(1) was reconstructed by the simulated spectrum uf(3). The results ate shown in Fig. 2. The fine
structure of the spectrum is reconstructed quite well for i."-40 and somewhat less well for i30. The latter is ex-
plained by the fact that the number of points (i.e., the number of analyzer channels), by which the experimental
spectrum is defined, is inadequate for describing it accurately, and consequently, the errors of integration with re-
spect to a small number of points give an excessively smooth curve in this region.
On the basis of calculations carried out, it can be concluded that the fine structure of the spectrum can be de-
termined not only by means of more detailed experiments but also by means of a more complete mathematical pro-
cessing of the measurement results. In this case, bearing in mind the subsequent mathematical processing, it is nec-
essary to improve some or other parameters of the instrument by the appropriate method, by selecting the optimum
relationships between them (in the given case these parameters are the speed of rotation of the rotor, the analyzer
channel width and the magnitude of the statistical errors).
750
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Calculation of the Scalar Energy Flux in a Moderator by the Measured Directional
Flux
In measurements of the energy spectrum of thermal neutrons in a reactor lattice the neutron beam, as a rule,
is extracted from a certain point of the lattice, parallel to the axis of the block. Thus, the energy spectrum of the
directional neutron flux is measured, which may be significantly different from the energy spectrum of the scalar
neutron flux and which, in practice, is the most interesting.
For a one-dimensional lattice with isotropic neutron scattering, the scalar energy flux cl)(E, r) at the point r is
related with the energy spectrum of neutrons in the direction of the lattice axis 4.11 (E, r) by the relationship
E JIM
s (E) Ea (E)] (RI (E, r) s (E, E', r) CD (E', r) dE' S (E, r)
(9)
(all the symbols here are universal).
If the directional flux 43.11 (E, r) at the point and the characteristics of the medium Ea (E), Es(E, E'), S(E, r),
are known, the scalar flux at the same point can be determined by solving Eq. (1). The method described above
enables a stable solution of this equation to be found.
The determination of the scalar flux in uranium-water lattices are of particular interest. In [4], the directional
flux was determined for these lattices by calculating the scalar flux and it was compared with the measured flux.
It was shown in this paper that the calculated spectra are slightly sensitive to scattering anisotropy in water. Thus,
in order to find the scalar flux in water by the measured directional flux, Eq. (9) can be used.
We shall consider by way of example, a uranium-water lattice for which the energy spectra of the directional
flux were measured in [2] at different points of the cell.
The differential scattering cross section of neutrons in water was calculated by the UPRAS (Universal Program
for Calculation of Cross sections) program on the assumption that water is described by a crystal model having an
acoustic spectrum which is determined experimentally [6], and an optical spectrum with frequencies w1 = 0.2 eV
and co2 = 0.4 eV. The calculated inelastic scattering cross sections agree well with the experimental data. The
thermal neutron sources were assumed to be isotropic and they were calculated on the assumption that for an energy
E > 0.25 eV the epithermal neutron spectrum has a Fermi form CE' and is independent of the angle. The quantity
C was defined as the limit of the experimental directional flux, multiplied by E [C = E (E)] at high energies.
Figure 4 shows the graph of the measured directional and calculated scalar neutron fluxes in water at the
boundary with the block.
In conclusion, it should be noted that as a result of reconstructing the true spectrum the errors, which distorted
the shape of the experimental curve, are increased generally speaking. The problem concerning the optimum choice
of the parameter a is associated with how much the measurement data, containing errors, permits the true spectrum
to be reconstructed. With too small a value of a, the errors in the reconstructed spectrum become so large that they
lead to buildup of the solution obtained. Paper [7] is devoted to the estimation of the errors in the reconstructed
spectrum. In addition, a special paper will be devoted to this.
LITERATURE CITED
1. A. N. Tikhonov, Dokl. AN SSSR, 149, 529 (1963).
2. V. L Mostovoi et al., Atomnaya Energiya, 13, 547 (1962).
3. V. I. Mostovoi, M. I. Pevzner, and A. I. Tsitivich, In the book: Data of the International Conference on the
Peaceful Uses of Atomic Energy, Geneva, 1955, [in Russian], Moscow,Izd-vo AN SSSR (1957), p. 19.
4. H. Honeck and H. Tacahashi, Nucl. Sci. Engng, 15, 115 (1963).
5. L. V. Mayorov, V. F. Turchin, and M. S. Yudkevich, Report No. 360, presented by USSR at the Third Inter-
national Conference on the Peaceful Uses of Atomic Energy [in Russian], Geneva (1964).
6. P. Egelstaff et al., Proc. Simp. Chalk-River, Vol. 1 (1962), p. 343.
7. A. N. Tikhonov and V. B. Glasko, Zh. vychislitel'noi i matematicheskoi fiziki, 4, 564 (1964).
751
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MEASUREMENT OF NEUTRON SPECTRA IN NICKEL,
IRON, AND STAINLESS STEEL
(UPC 621,039,538/539,125.52)
I. I. Bondarenko, V. , Liforov, V. N. Morozov,
M, N. Nikolaev, V. A. Parfenov, and V. A. Semenov
Translated from Atoninaya tnergiya, Vol, 18, No. 6,
pp. 593601, June, 1965'
Original article submitted July 13, 1964
Measurements of neutron spectra in various media, obtained by the transit-time method in the fast
pulse reactor IBR with a resolution of 0.04 Asec/m, are presented. The spectra of neutrons emerging
from iron and nickel prisms of various thickness and also those from stainless-steel prisms are studied.
The "fine structure" due to the resonance character of the cross sections of the media studied was
clearly seen in the spectra measured. The experimental neutron spectra are compared with calcu-
lations made with n multigroup system of constants allowing for resonance self-screening of the cross
sections. The reasons fpr the slight discrepancies found are analyzed.
The most widespread methods of measuring space-energy distributions of fast and intermediate neutrons in
various media are those based on measuring the spatial dependence of the rates of reactions possessing different
energy/cross section relationships [1-5]. Conclusions based on the results of such measurements are not always un-
equivocal, since at certain points of the set studied it is often necessary to determine the energy spectrum of the
neutrons with more precise (and complex) direct methods of neutron spectroscopy, by means of photoemulsion [1, 2],
the Wilson chamber [1], He3 chambers and counters [2, 3], or in some cases by means of resonance detectors [3-5].
All these methods, however, lack the resolving power needed to reveal the "fine structure" of the neutron
spectrum, associated with the resonance nature of the cross section, explicitly in the experimental results. At the
same time, a study of just this structure is extremely important for checking and developing ideas on the propaga-
tion of neutrons in media containing neutron-resonance scatterers and absorbers. I. I. Bondarenko first snowed the
necessity of considering the resonance structure of cross sections when calculating fast reactors in 1957. Since then
a great deal of work has been done on developing metnods of allowing for the resonance structure of cross sections
in reactor calculations [6-12], studying the influence of resonance effects in macroscopic experiments [13, 14], and
developing methods for determining the mean cross-section characteristics needed to allow for resonance effects in
reactor calculations [12, 15]. In all these papers, however, the influence of the resonance structure of the cross sec-
tions on the propagation of neutrons in matter was considered only from the point of view of determining integral
characteristics. Detailed neutron spectra in the resonance region were not examined.
At the present time, the only experimental method making it possible to measure neutron spectra with a high
resolving power is the transit-time method. Use of this method for studying the spectrum of fast neutrons retarded in
matter is complicated by the fact that the minimum pulse length is determined by the statistical indeterminacy of
the slowing -down time, which for nuclei with mass number ?50 is ?15 ?sec. Thus, for such investigations we need
a pulse source of high intensity, making it possible to increase the resolution on account of the greater transit-time
base. Owing to the absence of such a source, it appeared, until recently, impossible to use the transit-time method
for studying fast-neutron spectra.
Such a possibility re-emerged with the arrival of the fast pulse reactor IBR [16]. The pulse length of the IBR
reactor, equal to ?36 ?sec, is great enough for the slowing down of the neutrons in a heayy moderator not to lead to
serious prolongation of the pulse. At the same time, the high intensity of the pulse makes it possible to use an
752
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4\
3
I
111 Ilk?r?=-; --it
_
'L ?
E?
1
!002m
750 m
Cl'
Fig. 1. General arrangement of the experiment: 1) scintillation detector; 2) collimator; 3) boron-counter
detector; 4) monitor on 50-m base; 5) prism of material studied; 6) active zone of IBR reactor; 7) monitor
on 100-m base.
Fig. 2. General view of the disposition of the prism in the reactor room: 1) active zone of IBR reactor;
2) prism of material being studied; 3) platform for moving prism; 4) motor for moving platform; 5) neu-
tron guide; 6) slide; A) position of prism "at the zone;" B) position of prism "at the wall."
extremely large transit-time base (1 km) and thus, ensure a high resolving power of the system. It was, therefore,
decided to make a cycle of experiments to study the spectra of neutrons emerging from blocks of nickel, iron, and
stainless steel. The choice of materials was based on the following considerations.
As we know, the resonance structure of the nuclei of nickel and iron appears right up to energies of the order
of several MeV [4], and in the region of several keV it is extremely strong [6]. A characteristic feature of the
resonances in the keV range of energies is the presence of deep interference minima, which exert a very great in-
fluence of the diffusion and slowing-down of the neutrons [10]. The effect of interference between resonance and
potential scattering arises most strongly in the distribution of neutrons in media containing even-even nuclei (iron,
nickel), especially in those cases where one isotope predominates in the natural mixture (iron).
Study of the slowing-down spectrum of neutrons in stainless steel is interesting from the point of view of check-
ing the accuracy of existing methods of describing the propagation of neutrons in a mixture of resonance scatterers
[6]. In the choice of materials for study, of course, the important fact that iron, nickel, and stainless steel are widely
used in reactor construction was borne in mind.
753
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cP(f)
10
10
ba"
10
IP
tigs E, eV
Fig. 9, Spectrnm of leakage neutrons from nickel (curves 1, 2, 3, correspond to prism thicknesses of 31, 52, 77 cm).
The spectra of neutrons emergpg from nickel prisms 306, 515, and 770 mm thick, iron prisms 300, 480 mm
thick, and stainless7steel prisms 300 and 499 cm thick were measured. The transverse dimensions of the nickel
prisms were 800?. 1, it is possible to achieve stability of the bunch purely on account of the automatic
control system with respect to the center I ax , 13n1 both before and after the transition. The stability of the bunch
can be increased by introducing feedback via the dimensions. For other distributions, when there is natural damping,
it is necessary to analyze the parameter [5, 6] before and after the critical energy is reached, as done, for example,
in [6]. Owing to the non-linearity of the equations, the bunch's center does not coincide with the equilibrium phase,
but is displaced relative to it by an amount equal to the equilibrium dimensions of the bunch [5]. The center of the
bunch must be displaced. The results obtained are valid for bunches of small dimensions in the linear approximation.
To solve the non-linear problem, special discussion is required.
The author wishes to thank A. N. Lebedev for guidance, and A. A. Kolomenskii and g . L. Burshtein for dis-
cussing the results.
LITERATURE CITED
1. K. Johnsen and C. Schmelzer, Proc. of Int. Conf. on High Energy Accel. CERN (1956), p. 395.
2. W. Schnell, Proc. of Int. Conf. on High Energy Accel. CERN (1959), p. 485.
3. Yu. S. Ivanov and A. A. Kuz'min, Pribory i tekhnika eksperimenta, 4, 106 (1962).
4. H. Hereward, Proc. of Int. Conf. on High Energy Accel. Brookhaven (1961), p.236.
5. L. A. Zhil'kov and A. N. Lebedev, Atomnaya gnergiya, 18, 22 (1965).
6. g. A. Zhil'kov, Pribory i tekhnika eksperimenta, 1, 17 (1965).
7. A. A. Kolomenskii and A. N. Lebedev, Theory of Cyclic Accelerators [in Russian], Ch. IV, Moscow, Fizmatgiz
(1962).
8. g. A. Zhil'kov, Atomnaya gnergiya, 18, 58 (1965).
802
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Declassified and Approved For Release 2013/09/24: CIA-RDP10-02196R000700010006-1
PARTICLE LOSSES DUE TO PASSAGE THROUGH NONLINEAR
RESONANCES IN ACCELERATORS AND STORAGE DEVICES
(UDC 621.384.61)
A. A. Kolomenskii
Translated from Atomnaya Energiya, Vol. 18, No. 6,
PP- 636-638, June, 1965
Original article submitted July 3, 1964
Reports have recently appeared concerning projects for large proton storage systems, designed to effect colli-
sions between intersecting beams [1, 2]. Our aim in this paper is to analyze one of the possible mechanisms of par-
ticle losses which must be allowed for in developing and applying systems of this type.
A characteristic property of cyclic accelerators and storage rings is the existence of dangerous resonance com-
binations of the parameters, determined by the relations
q'2 == in ((h., z, in are integers),
(1)
where v r, v z = frequencies of radial and vertical betatron vibrations (expressed in terms of the frequency of revolu-
tion w). The resonance order k is determined by k = lq11 + lqz1. The permissible limits follow from the require-
ments that the working point v r, vz on the stability diagram shall not come close to the edges of the cell deter-
mined by the linear resonance bands (k :s 2). This cell is crossed by bands of non-linear resonance (k > 2), caused
by disturbances in the magnetic field. The effects of these resonances are usually not very dangerous, because when
the working point is stationary they are compensated for by non-linear displacement of the vibration frequency.
However, if the working point, though remaining within the cell, wanders and passes through non-linear resonance
bands owing to instabilities of the magnetic field, the amplitude of betatron vibrations may rise to dangerous values,
leading to loss of particles. There is especial danger that such losses may arise from repeated passage through non-
linear resonances when these occur in storage rings, in which the particles have to circulate for very long times of an
hour or more, several orders greater than those usually found in accelerators.
We shall confine ourselves to systems with strong focusing, having proton storage rings in mind. For electrons,
an important part is played by relativistic electromagnetic radiation, which (with suitable choice of focusing system)
damps the vibrations and limits the effects of various disturbing factors. For protons, this radiation is practically
nonexistent, and there is no damping of the vibrations, so that in storage rings the particle energy and magnetic field
remain practically constant.
If a nonlinear resonance is crossed slowly enough, there may be "coupling hysteresis" of the fraction of particles
in the resonance, accompanied by marked increase in the amplitude, even for a single non-repeated passage [3].
However, it is estimated that in practice passage is comparatively rapid. There is, thus, no coupling hysteresis, and
increases in amplitude will arise in those short periods during which there is synchronism between particle vibrations
and harmonic disturbances of the magnetic field.
The increases in amplitude due to rapid passage through non-linear resonances can be calculated from known
formulae (see [3], ch. III, /8).
az
It is characterized by the coefficient J = 2 , where ai, a are the amplitudes of vibration before and after trans-
mission, respectively. Using the method of stationary phases, we obtain for J the expression
2
J = [1 ? (? ? 1) G 25.1 sin (kivo
2 16'1
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(2)
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Here k = order of resonance (k > 2); coo = phase at moment of transmission;
G (uq, a, a) I
k
2pai 2
2R
where = averaged coefficients expressed in terms of floquet function f and proportional to the q-th
(resonance) harmonic of the disturbance (see Table 3 in [3]); p = momentum of particle; and R = radius of orbit.
d6
The expression 6' = ?0- occurring in (2) determines the rate of change of detuning, 6 = v -
the azimuthal angle, .9- . This can be expressed as follows:
Q AB
6 v
ct B
(3)
, in terms of
(4)
where ,A B, SZ = amplitude and frequency of disturbing oscillations in the magnetic field, the value of which at the
orbit is denoted by B. Using (4) we can easily estimate the rate of passage through a resonance and assure ourselves
that, in most practical cases, we can ignore "coupling hysteresis" in the resonance and consider the passage to be
rapid, satisfying (8.5) of [3] (ch. III, E8).
To simplify the calculation, we shall restrict ourselves to a cubic resonance, k = 3, V = q/3. By (2),(3) and
the above-mentioned table in [3], we have
J [
? 163,1 16,211111 11/.3- ?2
rb
21 I sin (300 1-2
4)]
(5)
where bq is the q-th harmonic of azimuthal asymmetry of the field. For the non-linearity characteristic of an ideal
field we can introduce the value
Bh B
- 1)h
ok
(6)
which is a generalization of the ordinary field index ri = no. In strong-focusing machines, nk usually obeys
1 < I vol< ni I .