Influence of second oder aberrations on mass-separation with 270° analyzing magnet
In this paper we study nonlinear characteristics of the wide aperture dipole magnet CP-17, which is utilized as an analyzing magnet in the stable isotope separator of the INR (Institute for Nuclear Research) in Kiev. A brief theory of the second order aberration of the magnetic field is outlined. Ba...
Saved in:
| Date: | 2008 |
|---|---|
| Main Authors: | , , , , |
| Format: | Article |
| Language: | English |
| Published: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2008
|
| Series: | Вопросы атомной науки и техники |
| Subjects: | |
| Online Access: | http://dspace.nbuv.gov.ua/handle/123456789/111497 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Influence of second oder aberrations on mass-separation with 270° analyzing magnet / A. Dolinskii, M. Dolinska, A. Valkov, A. Gorda, T. Rudenko // Вопросы атомной науки и техники. — 2008. — № 5. — С. 77-80. — Бібліогр.: 3 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-111497 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1114972017-01-11T03:03:05Z Influence of second oder aberrations on mass-separation with 270° analyzing magnet Dolinskii, A. Dolinska, M. Valkov, A. Gorda, A. Rudenko, T. Физика и техника ускорителей In this paper we study nonlinear characteristics of the wide aperture dipole magnet CP-17, which is utilized as an analyzing magnet in the stable isotope separator of the INR (Institute for Nuclear Research) in Kiev. A brief theory of the second order aberration of the magnetic field is outlined. Based on this theory numerical calculations have been performed to study the achievable power resolution depending on nonlinear magnetic field and on beam parameters. Досліджуються характеристики широко-апертурного магніту СП-17, який планується використовувати як аналізуючий магніт мас-сепаратора ІЯД (Інститут ядерних досліджень) у Києві. Наведена коротка теорія аберацій другого порядку. На основі описаної теорії проведено цілий ряд чисельних розрахунків для дослідження розподільної здатності сепаратора в залежності від нелінійності магнітного поля, а також від параметрів пучка. Исследуются характеристики широко-апертурного дипольного магнита СП-17, который планируется использовать в качестве анализирующего магнита масс-сепаратора ИЯИ (Институт ядерных исследований) в Киеве. Приведена краткая теория аберраций второго порядка. На основе описанной теории проведена серия численных расчетов для исследования разрешающей способности сепаратора в зависимости от нелинейности магнитного поля, а также от параметров пучка. 2008 Article Influence of second oder aberrations on mass-separation with 270° analyzing magnet / A. Dolinskii, M. Dolinska, A. Valkov, A. Gorda, T. Rudenko // Вопросы атомной науки и техники. — 2008. — № 5. — С. 77-80. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS 41.85.Gy http://dspace.nbuv.gov.ua/handle/123456789/111497 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Физика и техника ускорителей Физика и техника ускорителей |
| spellingShingle |
Физика и техника ускорителей Физика и техника ускорителей Dolinskii, A. Dolinska, M. Valkov, A. Gorda, A. Rudenko, T. Influence of second oder aberrations on mass-separation with 270° analyzing magnet Вопросы атомной науки и техники |
| description |
In this paper we study nonlinear characteristics of the wide aperture dipole magnet CP-17, which is utilized as an analyzing magnet in the stable isotope separator of the INR (Institute for Nuclear Research) in Kiev. A brief theory of the second order aberration of the magnetic field is outlined. Based on this theory numerical calculations have been performed to study the achievable power resolution depending on nonlinear magnetic field and on beam parameters. |
| format |
Article |
| author |
Dolinskii, A. Dolinska, M. Valkov, A. Gorda, A. Rudenko, T. |
| author_facet |
Dolinskii, A. Dolinska, M. Valkov, A. Gorda, A. Rudenko, T. |
| author_sort |
Dolinskii, A. |
| title |
Influence of second oder aberrations on mass-separation with 270° analyzing magnet |
| title_short |
Influence of second oder aberrations on mass-separation with 270° analyzing magnet |
| title_full |
Influence of second oder aberrations on mass-separation with 270° analyzing magnet |
| title_fullStr |
Influence of second oder aberrations on mass-separation with 270° analyzing magnet |
| title_full_unstemmed |
Influence of second oder aberrations on mass-separation with 270° analyzing magnet |
| title_sort |
influence of second oder aberrations on mass-separation with 270° analyzing magnet |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2008 |
| topic_facet |
Физика и техника ускорителей |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/111497 |
| citation_txt |
Influence of second oder aberrations on mass-separation with 270° analyzing magnet / A. Dolinskii, M. Dolinska, A. Valkov, A. Gorda, T. Rudenko // Вопросы атомной науки и техники. — 2008. — № 5. — С. 77-80. — Бібліогр.: 3 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT dolinskiia influenceofsecondoderaberrationsonmassseparationwith270analyzingmagnet AT dolinskam influenceofsecondoderaberrationsonmassseparationwith270analyzingmagnet AT valkova influenceofsecondoderaberrationsonmassseparationwith270analyzingmagnet AT gordaa influenceofsecondoderaberrationsonmassseparationwith270analyzingmagnet AT rudenkot influenceofsecondoderaberrationsonmassseparationwith270analyzingmagnet |
| first_indexed |
2025-07-08T02:14:57Z |
| last_indexed |
2025-07-08T02:14:57Z |
| _version_ |
1837043178773938176 |
| fulltext |
INFLUENCE OF SECOND ODER ABERRATIONS ON MASS-
SEPARATION WITH 270° ANALYZING MAGNET
A. Dolinskii1,2, M. Dolinska1, A. Valkov1, A. Gorda2, T. Rudenko1
1Institute for Nuclear Research NAS Ukraine, Kyiv, Ukraine
2GSI, Darmstadt, Germany
E-mail: A. Dolinskii@gsi.de, fax: 044-525-44-63, tel. 044-525-23-49
In this paper we study nonlinear characteristics of the wide aperture dipole magnet CP-17, which is utilized as an
analyzing magnet in the stable isotope separator of the INR (Institute for Nuclear Research) in Kiev. A brief theory
of the second order aberration of the magnetic field is outlined. Based on this theory numerical calculations have
been performed to study the achievable power resolution depending on nonlinear magnetic field and on beam pa-
rameters.
PACS 41.85.Gy
1. INTRODUCTION
For each isotope separator the goal is to achieve high
power resolution and at the same time large transmis-
sion as well as large momentum acceptances. However,
the achievable power resolution of the isotope separator
is limited by the many reasons, including second order
aberrations.
Usually the magnetic field of any magnetic-optical
devices is not ideal because of some constructive inac-
curacies in configuration of poles, some errors in coils
installation and etc. Therefore each magnetic optical
element produces aberrations – image distortion, which
has the effect of increasing image sizes on a focal plane
that limits the achievable resolution. This effect is called
“emittance blow-up”. Uncorrected aberrations may in-
crease the image size by a factor of five or more with a
proportional reduction in mass resolution. It is known
that in a linear approximation a maximum attainable
resolution is determined by the emittance of an ion
beam coming out an ion source and dispersion of the
isotope separator on the focal plane.
The mass resolution P is defined by formula
____________________________________________________________
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2008. № 5.
Series: Nuclear Physics Investigations (50), p.77-80.
77
m x DP
m x x
Δ
= =
Δ δ δ
, (1)
where Δx is the separation between two beams of mass
m and δx is the width of full image at the final
focus, D is the dispersion.
m+ Δm,
In this paper we study the influence of aberrations
on the resolution of a mass separator, which contains
only one large aperture dipole magnet CP-17 [1, 2].
This magnet has an inhomogeneous magnet field with
straight pole boundaries. The dispersion of the separator
with such magnet varies depending on the optical setup
of the system. There is condition, when the maximum
value of D can be reached 11 m. For example, if im-
age’s width is 1 mm the maximum possible resolution
of the separator is 11000. With such power resolution
any isotopes can be separated with low contaminations
of undesired ions e. g. with high enrichments. But dif-
ferent types of nonlinear field errors reduce resolving
power of the separator. To find out the nonlinear fea-
tures of the large aperture magnet we consider only sec-
ond order aberrations, which are stronger in comparison
with the higher order field imperfections. The numerical
algorithm based on the theory [3] and applied in our
code is briefly described.
In general, it is not hard to obtain the solution of
second-order equation by hand. But this solution is
rather complicated to study the particle beam behaviour
in optical system, which consists of several magnets.
The Monte-Carlo method is one of the tools that gives a
simple way to investigate the ion beam characteristics in
complicated ion-optical systems.
2. SOURCES OF MAGNETIC FIELD NON-
LINEARITIES
Particle motion in the magnetic field can be de-
scribed by equations including second order terms [3]:
;5.0)2(5.0
)2(5.0
)12()1(
22233
22
232
δβ
δ
βδ
hyhyyhyhnhh
xhnxxxh
xhnhxhnx
−′−′′++−′′
+−+′+′′
+−−+=−+′′
(2)
2 3
2
2( )
,
y'' nh y n h xy h'xy'
hx'y' nh y
+ = β − + −
+ δ
(3)
where
22
2
000
00
1;
2
y y
xx y
ye
B B
n
B Bx x ==
==
⎡ ⎤ ⎛∂ ∂ρ
β = = −ρ⎜ ⎟⎢ ⎥ ⎜ ⎟∂∂⎢ ⎥ ⎝ ⎠⎣ ⎦
⎞
,
n is the magnetic field index, is the bend-
ing radius and
Δ / ,δ = ρ ρ ρ
/ .h = 1 ρ
The general analytical solution of equations (2) and
(3) is presented as combination of the linear and nonlin-
ear second order terms of the Taylor's series:
( )0 0 0 0| | | Δ ,abx x x x x x' x' x x= 〈 〉 + 〈 〉 + δ δ + (4)
0 0 0 0| | Δ ,aby y y y y y' y' y= 〈 〉 + 〈 〉 + (5)
Δ abx and Δ free the non linear terms expressed by aby
2 2
0 0 0 0 0 0 0
2 2 2
0 0 0
2 2 2 2
0 0 0 0 0 0 0 0
Δ | | |
| | |
| | | ;
abx x x x x x x' x x' x x x
x x' x x' x' x
x y y x y y' y y' x y' y'
= 〈 〉 + 〈 〉 + 〈 δ〉
+〈 〉 + 〈 δ〉 δ + 〈 δ 〉δ +
〈 〉 + 〈 〉 + 〈 〉
δ
(6)
0 0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0
Δ | |
| |
| | .
aby y y x y x y x y' y'
y x' y y x' y' x' y'
y y y y y' y'
= 〈 〉 + 〈 〉
〈 〉 + 〈 〉 +
〈 δ〉 δ + 〈 δ〉 δ
+
(7)
The terms noted by bracket <> are unknown coeffi-
cients, which have to be defined. For convenience
mailto:Dolinskii@gsi.de
All of these terms will be marked as Substitution
equations. (4) and (5) in equations. (2) and (3) we can
get the differential equations for coefficients
.ijq
:ijq
78
2 ,( )
2
,
2 2 2
( ),
1 ,
ij x y ij ij
x y
q'' q f s
n h nh
+ω =
ω = − ω =
(8)
( )ijf s are the excitation functions, which are given in
Appendix for corresponding coefficients The solu-
tion of equations (8) can be expressed as
.ijq
0
0 0
0
( ) ( )( ) ( )
( )
( ) ( )( ) ( ) ( ) ,
( )
k
k
s
k k
s
k k
f Cq s S s
W
f SC s C s q S s q'
W
τ τ ∂τ
= −
τ
τ τ ∂τ
+ +
τ
∫
∫ k
s
s h
.+
(9)
where T is the
transfer matrix of an ion-optical system, ,
are the principle solutions,
Using equation (9) and are defined by
( ) ( ) ( ) ( ) ( ) det 1,W C S' C' S Tτ = τ τ − τ τ = =
cos( )xC k= ⋅
sin( )xS k= ⋅ 2 (1 ) .xk n= −
q' q''
(10) 0
0 0
0
( ) ( ) ( ) ( )
;
k xS S
q' S' f C d C' f S d C'q
S'q'
= τ τ τ − τ τ τ +
+
∫ ∫
(11) 0 0
2
0 0
( ) ( ) ( ) ( )
( ) ( )
k kS S
k
q'' S'' f C d C'' f S d
S'C C'S f S C''q S''q' f
= τ τ τ − τ τ τ +
− + + = ω
∫ ∫
Thus the quadratic terms are expressed through the
linear solutions C, S (characteristic functions of the
main trajectories) and the corresponding excitation
functions ( ).igf S Using standard numerical method for
integrating of equations (9, 10) one can calculate the
particle trajectories in a first and second order field im-
perfections in any ion-optical system.
3. MASS-SEPARATOR IN FIRST ORDER
APPROXIMATION
To study influence of aberrations on the mass sepa-
ration we consider a separator, where the main compo-
nent is only one large aperture dipole magnet CP-17
with an inhomogeneous field [1]. Because there are no
any additional ion-optical elements, this magnet deter-
mines most of the beam parameters in the separator.
Such scheme gives us possibility to find out the contri-
bution of nonlinear filed in effect of the power resolu-
tion restriction. The CP-17 magnet has a 2700 bending
angle and axial radius of 2 m. This magnet has possi-
bility to produce a double focusing in both horizontal
and vertical directions by appropriate adjustment of the
quadrupole component of the magnetic field. It provides
an adjustable field index in range from zero to one.
This adjustment can be easily obtained by using the
special magnet coil. If is closed to 0.831 the mass
dispersion is equal to 11 m in the focal plane,
which is placed from the magnet exit by distance
3.36 m. For such value of the dispersion two beams
with mass numbers and are separated by
23 cm in the focal plane. To have simultaneous focusing
in both horizontal and vertical directions the distance
between the exit slit of the ion source and magnet en-
trance must be 3.36 m and in this case the focal plane
would be placed from magnet exit by 3.36 m. Fig.1
shows the particles trajectories calculated in a linear
approximation throughout the whole system in the hori-
zontal and vertical planes.
ρ
n
n
mD
mD
50A = 51A =
Fig.1. Particle trajectories throughout the separator
(a first order approximation, s1=s2=3.36 m)
In this case the resolution power is defined by the
ion beam emittance coming out from the ion source.
Fig.5 shows the curves that correspond to numerically
calculated dependence the resolution power P on the
beam emittance in a first order approximation. One can
note that P can be larger than 1000 for emittance up to
10mm mrad⋅ and the momentum spread less than 104. It
should be noted that here and in all following simula-
tions the values of the emittance are statistical val-
ues with standard deviation 4. It is necessary to ensure
situation when the overlap between different isotopes
would be less than 0.01%, since the beam contains
99.99% of the particles at (usually emittance
calculations use
rms
4 rmsδ
2 rms,δ 95% of the beam).
The influence of aberrations brings an additional up-
per limit on obtaining resolution power. Using the-
ory given in Section 2 series of calculations has been
performed in a second order approximation.
4. SECOND ORDER PARTICLE TRACKING
The aberrations of the magnetic field reduce the
mass resolution of separator due to increasing of beam
spots in the focal plane. Mainly beam distortion is
caused by the sextupole components 2 / 2H x∂ ∂ of the
magnetic field, which create the excitation functions
.ijf However, the experimentally measured magnetic
field map is not always known. In this case one needs to
make some assumptions based on a theoretical calcula-
tion. For rough estimation of sextupole gradient
we can use the formula 2 /d H dx2
2
2
2
1 ,
2
y
S
H
B
x
∂
δ =
∂
a (12)
where is a half-aperture of the magnet, is the
field error, which can be expressed through the parame-
ter of (magnet field accuracy)
a yBδ
Δ
0 Δ,sB Hδ = ⋅ (13)
0H is the main magnet field in the median plane. Using
equations (12) and (13) one can derive the expression
for the sextupole gradient:
2
0
2 2
2 Δ
.yH H
x a
∂
=
∂
(14)
As shown in [1] the theoretically calculated value
is about 0.75%. But in practice this value can be larger
by factor 2 and more especially for low field, at which
magnet should operate. For insurance in our calculations
we assume that is 2%.
Δ
Δ
Under such value Δ aberrations give significant
contribution to growth of the beam spot on the focal
plane compare to the first order approximation. Fig.2,a
shows how the beam spot on the focal is disturbed in
case of the second order field errors. The tail of beam
intensity distribution (Fig.3) appears due to particles
traveling through the magnet with high amplitude,
where the field nonlinearity is stronger. One should also
note that the influence of the nonlinear field effect is
amplified if the particles have relatively large momen-
tum deviations. Therefore the particle amplitude trajec-
tories are increased and the beam spot on focal plane is
blow up proportionally to the momentum spread of the
beam as shown in Fig.2,b. In this case the reduction of
the mass resolution has strong dependence on the mo-
mentum spread of the isotope beam.
Fig.2. Beam spots at the focal plane a. dρ=0;
b. ∂p/p=0,2% εx =10 mm⋅mrad
Fig.3. Horizontal beam profiles on the focal plane.
These distributions correspond to the particles beam
spots shown in Fig.2
In Fig.4 we present the numerically simulated de-
pendence of the mass resolution power on the momen-
tum spread in cases of both the first and second order
approximation. We see that the momentum spread gives
main contribution to the reduction of the mass resolu-
tion of the separator. For momentum spread less than
0.1% the second order aberrations is significant. The
results outcomes in Fig.4 eligible only in case the emit-
tance value of 10 mm·mrad.
Fig.4. Dependence of the resolution power on the
momentum spread 10 mm·mrad xε =
To see how the resolution power depends only on
emittance a series simulations have been done, where
momentum spread of beam is fixed at value of 0.1%.
The results of simulations are shown in Fig.5. Here we
see that second order aberrations have very strong effect
for emittances larger 10 mm·mrad. The power resolu-
tion catastrophically is dropped up to several units for
relatively large emittances 20…100 mm·mrad. However
situation can essentially be improved by means of a
sextupolar correction. Using a sextupole magnet posi-
tion of which is shown in Fig.1 one can significantly
compensate the influence of second order field imper-
fection but not completely (Fig.5).
Finally in Fig.6 we present achievable parameters of
ion beam, which should be supplied at obtaining de-
signed power resolutions. The results shown in Fig.6
were obtained after the particle tracking throughout the
second order magnet system, where the parameter Δ has
moderate lower value 0.5%.
Fig.5. Resolution power versus horizontal emittance
obtained with first and second order calculations
Δρ/ρ= 0.1%
CONCLUSIONS
The ion optic of a mass separator with single dipole
magnet CP-017 was studied by taking into account sec-
ond order aberrations. It was shown that the image aber-
ration due to the dipole magnet limits the power resolu-
tion of the separator if the relative magnet error field is
larger than 0.5%. The energy spread and the ion beam
emittance coming out of the ion source are another limi-
tations.
79
APPENDIX The results of the second order simulations are that
the resolving power is reduced by factor larger than 5
for emittance larger 20 mm·mrad if momentum spread
of beam does not exceed 0.1%. The main effect, which
appears due to second order aberrations, is long tail
formed in the particle distribution on the focal plane.
The way to prove the resolving power close to designed
values is to correct the particle trajectories coming out
the exit of the magnet by sextupole magnet. Then the
resolving power of 150 could be ensured for emittances
up to 70 mm·mrad at the momentum spread less 0.1%.
The excitation functions ijf (S) corresponding to the sec-
ond order coefficients ijq
ijq ( )ijf S
ψ | δx= 0H
2
0|x x 3 2 2( ) 0.5 ( )x xbh C s hC' s+
x 0 0| x x' 3 22 ( ) ( ) ( ) ( )x x x xbh C s S s hC' s S' s+
80
Fig.6. Emittance versus momentum spread for different
resolution power calculated in second order approxima-
tion, where Δ= 0.5%
REFERENCES
1. Ю.Г. Басаргин. Магнитная система большого
спектрометра заряженных частиц // ЖТФ. 1969,
т.39, в.8, с.1472.
2. A.E. Valkov, et al. Using of monochromator magnet
of cyclotron U-240 for stable isotope separation //
KINR scientific work degist. Kyiv, 2005, №1 (13).
3. V. Kotov, V. Miller. High energy particles focusing
and mass-separation. Moskow: “Atomizdat”, 1969.
0| δx 2 3( 2) ( ) 2 ( )ψ (s) + ( )ψ ( )x x x x xn h C s bh C s hC' s ' s x − − +
2
0|x x' 3 2 2( ) 0.5 ( )x xbh S s hS' s+
x 0| δx' 2 3( 2) ( ) 2 ( )ψ ( ) + ( )ψ ( )x x x x xn h S s bh S s s hS' s ' s− − +
2| δx 2 3 2 2
x x( 2) ψ ( ) ψ ( ) + 0.5 ψ ( )xn h s bh s h ' s− − +
2
0|x y 3 2 20.5(2η ) ( ) ( ) 0.5 ( )y y xn h C s S s hC' s− −
x 0 0| y y' 3 2(2η ) ( ) ( ) 0.5 ( )y y xn h C s S s hS' s− −
2
0|x y' 3 2 20.5(2η ) ( ) 0.5 ( )y xn h S s hS' s− −
0 0|y x y ( ) ( ) ( ) ( )x y x ytC s C s hC' s C' s +
0 0|y y x' ( ) ( ) ( ) ( )x y x ytC s S s hC' s S' s +
0 0|y x' y' ( ) ( ) ( ) ( )x y x ytS s C s hS' s C' s +
0| δy y ( ) ( ) ( ) ( )x y x ytS s S s hS' s S' s +
0| δy y' 2 ( ) ( )ψ ( ) ( )ψ ( )y y x y xnh C s tC s s hC' s ' s+ +
0| δy y' 2 ( ) ( )ψ ( ) ( )ψ ( )y y x y xnh S s tS s s hS' s ' s+ +
3 2 2
02 1 ; 2( ) ; ( / )ּ( /magn yb n t n h R H H x2 )= − −η = − −η η = ∂ ∂
Статья поступила в редакцию 08.01.2008 г.
ВЛИЯНИЕ АБЕРРАЦИЙ ВТОРОГО ПОРЯДКА НА МАСС-СЕПАРАЦИЮ С ПОМОЩЬЮ
270°АНАЛИЗИРУЮЩЕГО МАГНИТА
А. Долинский, М. Долинская, А. Вальков, А. Горда, Т. Руденко
Исследуются характеристики широко-апертурного дипольного магнита СП-17, который планируется
использовать в качестве анализирующего магнита масс-сепаратора ИЯИ (Институт ядерных исследований)
в Киеве. Приведена краткая теория аберраций второго порядка. На основе описанной теории проведена
серия численных расчетов для исследования разрешающей способности сепаратора в зависимости от
нелинейности магнитного поля, а также от параметров пучка.
ВПЛИВ АБЕРАЦІЙ ДРУГОГО ПОРЯДКУ НА МАС-СЕПАРАЦІЮ ЗА ДОПОМОГОЮ
270°АНАЛІЗУЮЧОГО МАГНІТУ
О. Долінський, М. Долінська, О. Вальков, О. Горда, Т. Руденко
Досліджуються характеристики широко-апертурного магніту СП-17, який планується використовувати
як аналізуючий магніт мас-сепаратора ІЯД (Інститут ядерних досліджень) у Києві. Наведена коротка теорія
аберацій другого порядку. На основі описаної теорії проведено цілий ряд чисельних розрахунків для
дослідження розподільної здатності сепаратора в залежності від нелінійності магнітного поля, а також від
параметрів пучка.
CONCLUSIONS
REFERENCES
APPENDIX
ВПЛИВ АБЕРАЦІЙ ДРУГОГО ПОРЯДКУ НА МАС-СЕПАРАЦІЮ ЗА ДОПОМОГОЮ 270(АНАЛІЗУЮЧОГО МАГНІТУ
О. Долінський, М. Долінська, О. Вальков, О. Горда, Т. Руденко
|