Transportation channels calculation method in MATLAB
Output devices and charged particles transport channels are necessary components of any modern particle accelerator. They differ both in sizes and in terms of focusing elements depending on particle accelerator type and its destination. A package of transport line designing codes for magnet optica...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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| Цитувати: | Transportation channels calculation method in MATLAB / G.P. Averyanov, Yu.A. Bashmakov, V.A. Budkin, V.V. Dmitrieva, I.O. Osadchuk // Вопросы атомной науки и техники. — 2014. — № 3. — С. 138-142. — Бібліогр.: 14 назв. — англ. |
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irk-123456789-802712015-04-15T03:02:27Z Transportation channels calculation method in MATLAB Averyanov, G.P. Bashmakov, Yu.A. Budkin, V.A. Dmitrieva, V.V. Osadchuk, I.O. Динамика пучков Output devices and charged particles transport channels are necessary components of any modern particle accelerator. They differ both in sizes and in terms of focusing elements depending on particle accelerator type and its destination. A package of transport line designing codes for magnet optical channels in MATLAB environment is presented in this report. Charged particles dynamics in a focusing channel can be studied easily by means of the matrix technique. MATLAB usage is convenient because its information objects are matrixes. MATLAB allows the use the modular principle to build the software package. Program blocks are small in size and easy to use. They can be executed separately or commonly. A set of codes has a user-friendly interface. Transport channel construction consists of focusing lenses (doublets and triplets). The main of the magneto-optical channel parameters are total length and lens position and parameters of the output beam in the phase space (channel acceptance, beam emittance − beam transverse dimensions, particles divergence and image stigmaticity). Choice of the channel operation parameters is based on the conditions for satisfying mutually competing demands. And therefore the channel parameters calculation is carried out by using the search engine optimization techniques. Представлен пакет прикладных программ по проектированию каналов транспортировки, предназначенный для разработки магнитооптических каналов в среде широко распространённой системы MATLAB. Поскольку динамику заряженных частиц в фокусирующих каналах удобно записывать в матричной форме, то использование системы MATLAB, в которой объектом обработки информации является матрица, вполне обосновано. Комплекс программ обладает наглядным и удобным интерфейсом. Канал транспортировки строится из фокусирующих систем − дублеты и триплеты. Основными характеристиками магнитооптических каналов являются их полная длина и места расположения линз, а также параметры пучка в выходной плоскости канала (аксептанс канала, эмиттанс пучка − поперечные размеры пучка, расходимость частиц и стигматичность изображения). Выбор рабочего варианта параметров канала производится из условия удовлетворения взаимно конкурирующих требований, и поэтому расчёт параметров канала осуществляется с использованием методов поисковой оптимизации. Представлено пакет програм з проектування каналів транспортування, призначений для розробки магнітооптичних каналів у середовищі широко поширеної системи MATLAB. Оскільки динаміку заряджених часток у фокусуючих каналах зручно записувати в матричній формі, то використання системи MATLAB, в якій об'єктом обробки інформації є матриця, цілком обґрунтоване. Комплекс програм має наочний і зручний інтерфейс. Канал транспортування будується з фокусуючих систем − дублети і триплети. Основними характеристиками магнітооптичних каналів є їх повна довжина і місця розташування лінз, а також параметри пучка у вихідній площині каналу (аксептанс каналу, емітанс пучка − поперечні розміри пучка, розбіжність часток і стигматичність зображення). Вибір робочого варіанта параметрів каналу робиться з умови задоволення взаємно конкуруючих вимог, і тому розрахунок параметрів каналу здійснюється з використанням методів пошукової оптимізації. 2014 Article Transportation channels calculation method in MATLAB / G.P. Averyanov, Yu.A. Bashmakov, V.A. Budkin, V.V. Dmitrieva, I.O. Osadchuk // Вопросы атомной науки и техники. — 2014. — № 3. — С. 138-142. — Бібліогр.: 14 назв. — англ. 1562-6016 PACS: 29.27.Eg http://dspace.nbuv.gov.ua/handle/123456789/80271 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 |
Динамика пучков Динамика пучков |
| spellingShingle |
Динамика пучков Динамика пучков Averyanov, G.P. Bashmakov, Yu.A. Budkin, V.A. Dmitrieva, V.V. Osadchuk, I.O. Transportation channels calculation method in MATLAB Вопросы атомной науки и техники |
| description |
Output devices and charged particles transport channels are necessary components of any modern particle accelerator.
They differ both in sizes and in terms of focusing elements depending on particle accelerator type and its
destination. A package of transport line designing codes for magnet optical channels in MATLAB environment is
presented in this report. Charged particles dynamics in a focusing channel can be studied easily by means of the
matrix technique. MATLAB usage is convenient because its information objects are matrixes. MATLAB allows the
use the modular principle to build the software package. Program blocks are small in size and easy to use. They can
be executed separately or commonly. A set of codes has a user-friendly interface. Transport channel construction
consists of focusing lenses (doublets and triplets). The main of the magneto-optical channel parameters are total
length and lens position and parameters of the output beam in the phase space (channel acceptance, beam emittance
− beam transverse dimensions, particles divergence and image stigmaticity). Choice of the channel operation parameters
is based on the conditions for satisfying mutually competing demands. And therefore the channel parameters
calculation is carried out by using the search engine optimization techniques. |
| format |
Article |
| author |
Averyanov, G.P. Bashmakov, Yu.A. Budkin, V.A. Dmitrieva, V.V. Osadchuk, I.O. |
| author_facet |
Averyanov, G.P. Bashmakov, Yu.A. Budkin, V.A. Dmitrieva, V.V. Osadchuk, I.O. |
| author_sort |
Averyanov, G.P. |
| title |
Transportation channels calculation method in MATLAB |
| title_short |
Transportation channels calculation method in MATLAB |
| title_full |
Transportation channels calculation method in MATLAB |
| title_fullStr |
Transportation channels calculation method in MATLAB |
| title_full_unstemmed |
Transportation channels calculation method in MATLAB |
| title_sort |
transportation channels calculation method in matlab |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2014 |
| topic_facet |
Динамика пучков |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/80271 |
| citation_txt |
Transportation channels calculation method in MATLAB / G.P. Averyanov, Yu.A. Bashmakov, V.A. Budkin, V.V. Dmitrieva, I.O. Osadchuk // Вопросы атомной науки и техники. — 2014. — № 3. — С. 138-142. — Бібліогр.: 14 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT averyanovgp transportationchannelscalculationmethodinmatlab AT bashmakovyua transportationchannelscalculationmethodinmatlab AT budkinva transportationchannelscalculationmethodinmatlab AT dmitrievavv transportationchannelscalculationmethodinmatlab AT osadchukio transportationchannelscalculationmethodinmatlab |
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| fulltext |
ISSN 1562-6016. ВАНТ. 2014. №3(91) 138
TRANSPORTATION CHANNELS CALCULATION METHOD
IN MATLAB
G.P. Averyanov1, Yu.A. Bashmakov1,2, V.A. Budkin1, V.V. Dmitrieva1, I.O. Osadchuk1
1National Research Nuclear University – Moscow Engineering Physics Institute, Moscow, Russia
E-mail: VABudkin@mephi.ru;
2P.N. Lebedev Physical Institute of RAS, Moscow, Russia
E-mail: bashm@x4u.lebedev.ru
Output devices and charged particles transport channels are necessary components of any modern particle accel-
erator. They differ both in sizes and in terms of focusing elements depending on particle accelerator type and its
destination. A package of transport line designing codes for magnet optical channels in MATLAB environment is
presented in this report. Charged particles dynamics in a focusing channel can be studied easily by means of the
matrix technique. MATLAB usage is convenient because its information objects are matrixes. MATLAB allows the
use the modular principle to build the software package. Program blocks are small in size and easy to use. They can
be executed separately or commonly. A set of codes has a user-friendly interface. Transport channel construction
consists of focusing lenses (doublets and triplets). The main of the magneto-optical channel parameters are total
length and lens position and parameters of the output beam in the phase space (channel acceptance, beam emittance
− beam transverse dimensions, particles divergence and image stigmaticity). Choice of the channel operation param-
eters is based on the conditions for satisfying mutually competing demands. And therefore the channel parameters
calculation is carried out by using the search engine optimization techniques.
PACS: 29.27.Eg
INTRODUCTION
Depending on the type of accelerator and its purpose
transport channels differ both on the length and on a set
of focusing elements whose parameters depend on the
type of accelerated particles and their energy. For parti-
cles with a momentum in the range from tens of MeV/c
to hundreds GeV/c basic channel elements are magnetic
quadrupole lenses and dipole bending magnets [1 - 9].
The main characteristics of the magneto-optical chan-
nels are full length of the channel, the location of the
lenses along the channel and magnitude of the magnetic
field in the lenses, as well as the acceptance of the
channel and the emittance of the beam at the output
plane of the channel (the transverse dimensions of the
beam and the angular divergence of a beam of particles)
and the stigmatic image [7]. Achievement of the re-
quired parameters is associated with the choice of a
compromise of mutually competing conditions imposed
on the parameters of the channel and thus final calcula-
tion of focusing system is made with search methods for
the minimum of function of many variables. The objec-
tive function, which determines the quality of the pro-
jected channel (ie, the degree of achievement of the set
of parameter values), is a multiextreme function, which
creates additional difficulties in the calculation of the
focusing system.
Existing software packages for magneto-optical
channels calculation (see for example [4, 6]), developed
and intended for the design of unique accelerator com-
plex are rather difficult to apply to the majority of mass-
produced accelerators for industrial and medical appli-
cations, as they require a lot of experience and sufficient
skills for their users. The proposed method is intended
to relatively simple designing of transport channels of
charged particles and allows a convenient and under-
standable form study the focusing process of focusing
charged particles by different systems and to assess the
degree of influence of each channel parameter on re-
quired sizes beam formation.
Since the dynamic of charged particles in the focus-
ing channel are conveniently written in matrix form, the
use of MATLAB, where information processing object
is a matrix, is quite justified. Program code is easy to
write and understand. Resources of the system allow use
the modular principle of the program creation. Software
modules of basic elements of magneto-optical systems
are small in size and easy to use. IFR «Katran» having
an opened modular structure can be adapted to the spe-
cific technical requirements of the accelerator installa-
tion. The developed method can also be used for train-
ing students in the course of CAD electric torch.
1. TRANSPORTATION CHANNELS
CALCULATION METHOD
1.1. MATHEMATICAL PROBLEM
As it is known the motion of a particle with charge е
in an electric field with a strength of E
and a magnetic
field with the induction of B
is described by the equation
])*[()( BvEevm
dt
d
+=
(1)
in which the right side is the Lorentz force, where m is
the total relativistic mass and 0 0 0r zz xx ss= + +
− radi-
us vector of the particle, v r=
is particle speed.
Fig. 1 shows a coordinate system where 1 and 2 - 3 -
4 - x and z are transverse coordinates, and s is longi-
tudinal coordinate along which particle motion per-
forms, 0 0 0, ,x z s is orts.
Fig. 1. Stationary Cartesian coordinate system {z, x, s}
mailto:VABudkin@mephi.ru
mailto:bashm@x4u.lebedev.ru
ISSN 1562-6016. ВАНТ. 2014. №3(91) 139
Taking into account that the energy of the relativistic
particles in a magnetic field does not change, and no
electric field, the equations of motion take the form
)( xs BsBx
m
ez −=
)( sz BzBs
m
ex −= (2)
)( zx BxBz
m
es −=
Going from time differentiation to differentiation
with respect to the longitudinal coordinate (s), making
the substitution variables and the corresponding trans-
formations, we obtain the equations of the particle tra-
jectory in the transverse planes [2].
]'')'1('[''1'' 222
zxs BzxBzBxxz
p
ez ++−++=
(3)
]'')'1('[''1'' 222
xzs BzxBxBzxz
p
ex ++−++−=
where p m= υ is momentum of the particle.
Transport channel is composed of quadrupole lenses
and drift spaces. In the most practically interesting cases
the transverse components of the magnetic field within
the lens can be considered as linear functions of the co-
ordinates, and the longitudinal component of the mag-
netic field is absent
,
,
0,
x
z
s
B gz
B gx
B
=
=
=
(4)
g is the magnetic field gradient.
Then the equations of motion (3) of the charged par-
ticles in the quadrupole take the form
( )
( )
2
2
2 2
2 2
1 1 .
1 1 ,
z k z x z z x z x
x k z x x x x z z
′′ ′ ′ ′ ′ ′= − + + + −
′′ ′ ′ ′ ′ ′= + + + −
(5)
where k characterizes the focusing magnetic field
strength ( * /k e g p= ).
The solutions of these equations ( )x s and ( )z s in
the region near the axis (axis s ) can be expanded in
powers of the initial parameters
( ) ( ) ( ) ( )0000 0000 xx,xx,zz,zz ′=′=′=′= .
In the practice in the majority cases it is enough only
to keep the linear terms of the expansion. Then ( )x s
and ( )z s are solutions of linear equations of the trajec-
tory (6) in the quadrupole
* 0,
* 0.
x k x
z k z
′′ − =
′′ + =
(6)
Solutions of the equations (6) in the matrix form on
the output of the lens with length l can be written as
'
0
0
' )()(*/
)(*/)(
z
z
chshl
shlch
z
z
θθθ
θθθ
= ,
'
0
0
' )cos()sin(*/
)sin(*/)cos(
x
x
l
l
x
x
θθθ
θθθ
−
= , (7)
where θ is transit angle. ( 3* / *g p lθ = ),
0 0 0 0, , ,x x z z′ ′ are the coordinates of the particle input,
, , ,x x z z′ ′ − coordinates of the particles at the output of
the channel section − linear and angular coordinates of
the particles relative to the optical axis of the channel.
The motion of a charged particle in free space in the
matrix form can be written as
'' z
zs
z
z
0
0
10
1
= '' x
xs
x
x
0
0
10
1
= , (8)
where s is the length of the drift space.
1.2. THE STRUCTURE OF THE RFP «KATRAN».
DESIGN STAGES TRANSPORTATION
CHANNEL
The basic flowchart of the software package and in-
terrelation of modules are given in Fig. 2. In the head
program key parameters of a beam of charged particles
and structurally set parameters of lenses and the channel
as a whole, and also values of target parameters neces-
sary for calculations, normalizing and weight coeffi-
cients are established.
Fig. 2. Block sheme of the modular structure
of the package
Design of the channel of transportation consists of
several stages. At the first stage the choice of "refer-
ence" option of a quadrupole lens, i.e. the lenses which
parameters allow to carry out in principle the necessary
focusing of a beam without claims for a system optimal-
ity as a whole is carried out. The subsequent construc-
tion and specification of focusing system can lead to
correction of parameters of a lens. At the second stage,
based on a "reference" version of the lens focusing sys-
tem formed in accordance with the requirements im-
posed on the transportation channel and conducted re-
search of their focusing properties based on a range of
characteristics. At the last stage, the selection of optimal
parameters focusing system and visualization of the
results − the construction of phase ellipses and the beam
envelope. Calculation of dynamics of particles is carried
140 ISSN 1562-6016. ВАНТ. 2014. №3(91)
out for a nominal momentum of charged particles
p = p0.
First step: Modules – «Elements channel» → «The
quadrupole lens» + «free period». The matrix of transi-
tion of quadrupole in focusing and defocusing planes
and the transition matrix of space are written down in
the modules. A pre-selection "reference" version of
quadrupole parameters − the magnetic field (B), the
length of the lens is carried out. The transition matrix
structure - the "free span - a lens" has the form
11 12 0
21 22 0
1
* * ,
0 1
m m xx F
m mx x
= ′ ′
′
=
′ 0
0
2221
1211
x
x
*
MM
MM
x
x
, (9)
′0
0
x
x is parameters of the beam at the entrance to the
structure,
′x
x is parameters of the beam at the output
of the structure,
2221
1211
mm
mm is transition matrix of lenses
1
0 1
F
− is transition matrix of free interval,
2221
1211
MM
MM
−
transition matrix structure of the "free period − lens".
Therefore, to determine the focal length of the lens F
necessary condition is 0** 220210 =′+=′ MxMxx ,
which is equal to
022 =M , because 00 =x . (10)
Performing matrix multiplication in (9) we obtain
22 21 22* / *sin * cos 0,M m F m l F= + = −θ θ + θ = (11)
22 21 22* / * * 0.N n F n l sh F ch= + = θ θ + θ = (12)
The focal lengths of the quadrupole lens in the fo-
cusing ( Ff ) and defocusing ( Fd ) planes.
/ * ,
/ * .
Ff l ctg
Fd l cth
= θ θ
= − θ θ
(13)
These relations allow for construction and analysis
of the range of characteristics of the quadrupole lenses.
Second stage: Modules − «Doublet» → «range of
characteristics." In modules are written procedures for
search focal length doublet and procedures for the for-
mation of the objective functions for different variants
of calculation. Based on the results of the first stage and
advancing relevant requirements of the numerical values
of the transition matrix elements in the transverse planes
can be formed by a beam of charged particles of the
desired configuration, and adjust the overall length of
the transport channel. As an example, consider the con-
struction of a system of two quadrupole lenses (dou-
blet), which provides stigmatic image of the output of
the system and tries to save the initial shape of the
beam.
Link between the output and the input parameters of
the beam is determined by the full transition matrix in
the area "source − the output of the second lens."
FODMF and DOFMF is the transition matrix of the
free period "source − the first lens",
D – the transition matrix of the free space between
the lens,
0
' '
0
2 * * 1 * * ,FOD xx
L D L MF
x x
=
011 12
' '
21 22 0
,
xx H H
H Hx x
=
(14)
0
' '
0
2 * * 1 * * ,DOF zz
L D L MF
z z
=
011 12
' '
21 22 0
.
zz V V
V Vz z
=
L1 and L2 are transition matrixes of the first and se-
cond elements in the appropriate orientation; H and V
are the transition matrixes in the area "source − the out-
put of the second lens" in the horizontal (FOD) and ver-
tical (DOF) planes.
If a point source is located on the optical axis by a
distance equal to the focal length of the doublet then
after doublet passing particles will travel on parallel
axis. Mathematical notation of this condition
022 =H and 22 0.V = (15)
Knowledge of focal lengths of the doublet FODF
and DOFF determines the geometry of the focusing
system. Fig. 3. Shows the dependence of focal length
doublet FODF and DOFF on the values of the transit
angle of the second lens at selected above "reference"
parameters of the first lens and a given distance between
the lenses. As seen from the graphs, there is a range of
values of the transit angle of the second lens, wherein
the focal length of doublet FODF and DOFF are posi-
tive. This is the "work area" suitable for further search-
es. To solve this problem we use extreme methods of
search and in this case, in objective function is setted
only one target parameter − 22 0H → (for FODF ) and
22 0V → (for DOFF ) by varying the values of the trans-
it angle of the second lens.
You can assess the nature of the influence of the dis-
tance between the lenses on the value of the "work area"
(see Fig. 3) − with increasing distance between the
lenses, "workspace" increases.
0.25 0.3 0.35 0.4
-100
-50
0
50
100
Ff
od
F
do
f ,
[m
]
teta2, [m]
d=0.20 0.30 0.40[m] t1=0.330 l=0.20[m]
P=0.10[GeV/c]
FOD
DOF
Fig. 3. Dependence of the local lengths of the doublet
transit angle of the second lens
The third stage: Modules − «stigmatic doublet» →
«transition matrices. The envelope of the beam »→«The
phase ellipse of the beam. The procedures for finding
ISSN 1562-6016. ВАНТ. 2014. №3(91) 141
optimal parameters of the doublet and procedures of the
objective function for the stigmatic image doublet, as
well as the procedure for constructing the phase ellipses
at the output of each element of the transport channel
and the construction of the beam envelope are written
down in the modules. For the quadrupole doublet lenses
are searched optimal values of the following parameters
− span angles of the two lenses, the distance from the
source to the first lens, the distance between the lenses
and the distance from the second lens to the image
plane. All these doublet parameters are included in the
formation of the elements objective function of the tran-
sition matrices channel.
2. EXAMPLE OF CALCULATING
OF TRANSPORT CHANNEL ELEMENTS
The mathematical formulation of the synthesis prob-
lem − to determine the optimal solution, which, for
whatever characteristics preferred over others solutions.
The problem is multiextremal because in the solutions
search area there are a number of local minima of the
objective function, which can be represented as
( )( )∑
=
−
=
5
1
2*
i
norm
i
ii
i
x
aFt
ϕ
ϕϕ
,
(16)
where ( )xiϕ − function depending on the vector (sys-
tem settings); *
iϕ − setting goals; norm
iϕ − normali-
zation factor; ai − weighting factor.
One of the features of the problem is bad scaling −
length of transport channel free gaps vary in the range
from tens of centimeters to hundreds of meters, while
the values of angles spans the lens lie in the range
0.2…0.5 rad. Scaling − transformation of the variables
with the appropriate normalization coefficients norm
iϕ
to this form when the value of the variables are compa-
rable in magnitude, and the objective function value is
measured in terms of dimensionless quantities are close
to unity.
As a method for finding the optimal solution uses
the internal functions of MATLAB realize the direct
search method (the simplex method) − the algorithm
Nelder-Milda and gradient methods.
In the context of solving the above problems as a
function ( )xiϕ are the transition matrices elements
22 0H = and 22 0V = to determine the focal length dou-
blet, 12 0HD = and 12 0VD = (full matrix elements of
doublet transmission) to obtain a stigmatic image, and
the total length of the focusing system. The weighting
factors take numerical values indicating the degree of
importance of the achievement of this component of the
objective function the parameter setting goals. Target
data *
iϕ − values set by the user.
In Table, the first column contains the parameters of
the starting vector, i.e. the value of the channel, which
begins the process of finding their optimum values. Pa-
rameters of starting vector are set based on analysis of
the first two stages of the calculation. These parameters
are described in the order of elements along the channel.
Symbols U, D, V − corresponding to free intervals
(measured in meters), θ − the transit angle lens (meas-
ured in radians).
From the analysis of wide-band characteristics of the
lens was chosen reference variant lens length L = 0.2 m,
with the induction of the magnetic field B = 1.6 kgs and
the transit angle θ = 0.42 when the momentum of the
particles P = 100 MeV/c. A result of optimization of
parameters (see Table) the total length of the canal will
14.5…20.5 m. These characteristics are consistent with
existing channels FIAN synchrotron [11].
Optimized parameters of the doublet a transition matrix
Starting
vector
Optimized
parameters
HD
Horizontal
plane
VD
Vertical
plane
U=10.0000
θ1=0.4300
D=0.3000
θ2=0.4500
V=10.0000
U=9.9935
θ1=0.4361
D=0.0911
θ2=0.4361
V=9.9919
-1.3137
0.0014
-0.2027
-0.7610
-0.7615
-0.0062
-0.2028
-1.3149
U=10.0000
θ1=0.4300
D=0.3000
θ2=0.4500
V=10.0000
U=9.4120
θ1= 0.4363
D=0.1068
θ2=0.4365
V=9.2326
-1.3087
0.0024
-0.2171
-0.7637
-0.7362
-0.0028
-0.2183
-1.3592
U=7.0000
θ1=0.4300
D=0.3000
θ2=0.4500
V=7.0000
U=6.9441
θ1=0.4104
D=0.2856
θ2=0.4104
V=6.9399
-1.4883
-0014
-0.2970
-0.6722
-0.6714
-0.0016
-0.2971
-1.4901
The first two solutions differ in the normalization
factor for the lengths of the input and output gaps. A
third option in the starting vector has a smaller dimen-
sion values of input and output space than the first em-
bodiment. The presented data show that the achieve-
ment of the optimal variant is mainly due to the selec-
tion of the values of the span angle and distance be-
tween the lenses. The third embodiment shows the
phase channel ellipses particle beam entering and leav-
ing the doublet (Fig. 4).
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
x',
z'[
m
ra
d]
x,z, [mm]
Fig. 4. Phase portrait of the beam: entrance plane –
a solid ellipse, output plane (horizontal) – dotted
ellipse; output plane (vertical) – pointed ellipse
In Fig. 5 the envelopes of the beam in the horizontal
(solid line) and vertical (dashed line) planes are shown
for symmetrical beam. Therefore at the initial part of the
channel from output of the accelerator to the first lens
envelopes on both planes coincide. If the maximum
angular particles divergence at the exit of the accelerator
142 ISSN 1562-6016. ВАНТ. 2014. №3(91)
is about 1 mrad then the maximum beam size at the
channel is ≈ 17 mm.
From these results one can conclude that for a given
channel geometry beam with a larger angular diver-
gence can be transported (up to 5 mrad with an aperture
of 10 cm), or it is possible to use a lens with a smaller
aperture (e.g., aperture 4 cm) or it is necessary to repeat
the search of focusing doublet parameters for satisfying
formulation problem. In such a sequence the design of
the quadrupole lenses triplet and symmetric cell is per-
formed.
0 5 10 15 20 25 30
-30
-20
-10
0
10
20
30
x,z
[
mm
]
s [m]
Q1 Q2
Xog
Zog
Fig. 5. Beam envelopes in horizontal (solid line)
and vertical (dotted line) plane
REFERENCES
1. E.D. Courant, M.S. Livingston, H.S. Snyder. The
strong-focusing synchrotron – a new high energy ac-
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2. K. Steffen. High-energy beams optics. М.: «Mir»,
1969.
3. V.A. Budkin, A.A. Glazkov. On the choice of optimal
optical channel of a separator by cybernetic method.
Accelerators. Iss. XII, М.: «Atomizdat», 1970.
4. TRANSPORT. A computer program designing
charged particle beam transport system. CERN 80-04.
5. K. Steffen. Basic course accelerator optics. CERN
85-19.
6. The MAD Program. CERN/LEP-TH/88-38.
7. P.W. Hawkes, E. Kasper, et al. Principles of elec-
tron optics. Elsevier, 1996.
8. V.I. Kotov, V.V. Miller. Focusing and separation on
the masses of the high-energy particles. M.: «Atom-
izdat», 1969.
9. V.A. Baskov, Yu.A. Bashmakov, A.V. Verdi, et al.
Diagnostics system of transport of an electron beam
on the basis of scintillation counters: Preprint FIAN,
1997, №13, p. 22.
10. F.P. Vasil’ev. Numerical methods for solving ex-
treme problems. M.: «Nauka», 1988.
11. G. Rekleytis, A. Reyvindarn, K. Regsdel. Optimiza-
tion in Engineering. Russian translation. In 2 books.
M.: «Mir», 1986.
12. E. Polak. Numerical optimization. Wiley, 1974.
13. G.P. Averianov, V.A. Budkin, V.V. Dmitrieva.
Computer-aided design. M.: «MEPhI», 2010.
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2011.
Article received 09.10.2013
МЕТОДИКА РАСЧЕТА КАНАЛОВ ТРАНСПОРТИРОВКИ ПУЧКОВ ЗАРЯЖЕННЫХ ЧАСТИЦ
В СРЕДЕ MATLAB
Г.П. Аверьянов, Ю.А. Башмаков, В.А. Будкин, В.В. Дмитриева, И.О. Осадчук
Представлен пакет прикладных программ по проектированию каналов транспортировки, предназначен-
ный для разработки магнитооптических каналов в среде широко распространённой системы MATLAB. По-
скольку динамику заряженных частиц в фокусирующих каналах удобно записывать в матричной форме, то
использование системы MATLAB, в которой объектом обработки информации является матрица, вполне
обосновано. Комплекс программ обладает наглядным и удобным интерфейсом. Канал транспортировки
строится из фокусирующих систем − дублеты и триплеты. Основными характеристиками магнитооптиче-
ских каналов являются их полная длина и места расположения линз, а также параметры пучка в выходной
плоскости канала (аксептанс канала, эмиттанс пучка − поперечные размеры пучка, расходимость частиц и
стигматичность изображения). Выбор рабочего варианта параметров канала производится из условия удо-
влетворения взаимно конкурирующих требований, и поэтому расчёт параметров канала осуществляется с
использованием методов поисковой оптимизации.
МЕТОДИКА РОЗРАХУНКУ КАНАЛІВ ТРАНСПОРТУВАННЯ ПУЧКІВ ЗАРЯДЖЕНИХ ЧАСТОК
У СЕРЕДОВИЩІ MATLAB
Г.П. Авер′янов, Ю.А. Башмаков, В.А. Будкін, В.В. Дмитрієва, І.О. Осадчук
Представлено пакет програм з проектування каналів транспортування, призначений для розробки магні-
тооптичних каналів у середовищі широко поширеної системи MATLAB. Оскільки динаміку заряджених
часток у фокусуючих каналах зручно записувати в матричній формі, то використання системи MATLAB, в
якій об'єктом обробки інформації є матриця, цілком обґрунтоване. Комплекс програм має наочний і зручний
інтерфейс. Канал транспортування будується з фокусуючих систем − дублети і триплети. Основними харак-
теристиками магнітооптичних каналів є їх повна довжина і місця розташування лінз, а також параметри пу-
чка у вихідній площині каналу (аксептанс каналу, емітанс пучка − поперечні розміри пучка, розбіжність час-
ток і стигматичність зображення). Вибір робочого варіанта параметрів каналу робиться з умови задоволення
взаємно конкуруючих вимог, і тому розрахунок параметрів каналу здійснюється з використанням методів
пошукової оптимізації.
INTRODUCTION
1. Transportation channels calculation method
1.1. mathematical problem
1.2. The structure of the RFP «Katran». Design stages transportation channel
2. Example of calculating of transport channel elements
REFERENCES
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