Spherical detector device mathematical modeling with taking into account detector module symmetry
Mathematical model for spherical detector device accounting to symmetry properties is considered. Exact algorithm for simulation of measurements procedure with multiple radiation sources developed. Modelling results are shown to have perfect agreement with calibration measurements
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2005
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| Series: | Вопросы атомной науки и техники |
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| Cite this: | Spherical detector device mathematical modeling with taking into account detector module symmetry / V.G. Batiy, D.V. Fedorchenko, I.M. Prokhorets, S.I. Prokhorets, M.A. Khazhmuradov // Вопросы атомной науки и техники. — 2005. — № 6. — С. 63-65. — англ. |
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irk-123456789-812302015-05-14T03:02:37Z Spherical detector device mathematical modeling with taking into account detector module symmetry Batiy, V.G. Fedorchenko, D.V. Prokhorets, I.M. Prokhorets, S.I. Khazhmuradov, M.A. Экспериментальные методы и обработка даных Mathematical model for spherical detector device accounting to symmetry properties is considered. Exact algorithm for simulation of measurements procedure with multiple radiation sources developed. Modelling results are shown to have perfect agreement with calibration measurements Розглянуто математичне моделювання шарового детектора з урахуванням властивостей симетрії. Представлено послідовний алгоритм моделювання процедури вимірювань за наявності декількох джерел випромінювання. Показано, що результати моделювання добре узгоджуються з калібрувальними вимірюваннями. Рассмотрено математическое моделирование шарового детектора с учетом свойств симметрии. Представлен последовательный алгоритм моделирования процедуры измерения при наличии нескольких источников излучения. Показано, что результаты моделирования имеют хорошее согласие с калибровочными измерениями. 2005 Article Spherical detector device mathematical modeling with taking into account detector module symmetry / V.G. Batiy, D.V. Fedorchenko, I.M. Prokhorets, S.I. Prokhorets, M.A. Khazhmuradov // Вопросы атомной науки и техники. — 2005. — № 6. — С. 63-65. — англ. 1562-6016 PACS: 28.41Te http://dspace.nbuv.gov.ua/handle/123456789/81230 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Экспериментальные методы и обработка даных Экспериментальные методы и обработка даных |
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Экспериментальные методы и обработка даных Экспериментальные методы и обработка даных Batiy, V.G. Fedorchenko, D.V. Prokhorets, I.M. Prokhorets, S.I. Khazhmuradov, M.A. Spherical detector device mathematical modeling with taking into account detector module symmetry Вопросы атомной науки и техники |
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Mathematical model for spherical detector device accounting to symmetry properties is considered. Exact
algorithm for simulation of measurements procedure with multiple radiation sources developed. Modelling results
are shown to have perfect agreement with calibration measurements |
| format |
Article |
| author |
Batiy, V.G. Fedorchenko, D.V. Prokhorets, I.M. Prokhorets, S.I. Khazhmuradov, M.A. |
| author_facet |
Batiy, V.G. Fedorchenko, D.V. Prokhorets, I.M. Prokhorets, S.I. Khazhmuradov, M.A. |
| author_sort |
Batiy, V.G. |
| title |
Spherical detector device mathematical modeling with taking into account detector module symmetry |
| title_short |
Spherical detector device mathematical modeling with taking into account detector module symmetry |
| title_full |
Spherical detector device mathematical modeling with taking into account detector module symmetry |
| title_fullStr |
Spherical detector device mathematical modeling with taking into account detector module symmetry |
| title_full_unstemmed |
Spherical detector device mathematical modeling with taking into account detector module symmetry |
| title_sort |
spherical detector device mathematical modeling with taking into account detector module symmetry |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2005 |
| topic_facet |
Экспериментальные методы и обработка даных |
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http://dspace.nbuv.gov.ua/handle/123456789/81230 |
| citation_txt |
Spherical detector device mathematical modeling with taking into account detector module symmetry / V.G. Batiy, D.V. Fedorchenko, I.M. Prokhorets, S.I. Prokhorets, M.A. Khazhmuradov // Вопросы атомной науки и техники. — 2005. — № 6. — С. 63-65. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
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2025-07-06T05:42:00Z |
| last_indexed |
2025-07-06T05:42:00Z |
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1836875016633843712 |
| fulltext |
SPHERICAL DETECTOR DEVICE MATHEMATICAL MODELING
WITH TAKING INTO ACCOUNT DETECTOR MODULE SYMMETRY
V.G. Batiy1, D.V. Fedorchenko1, I.M. Prokhorets2, S.I. Prokhorets2, M.A. Khazhmuradov2
1Institute for Safety Problems of Nuclear Power Plants, Chernobyl, Ukraine
e-mail: oitp@mntc.org.ua
2National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
e-mail: khazhm@kipt.kharkov.ua, iprokhorets@kipt.kharkov.ua
Mathematical model for spherical detector device accounting to symmetry properties is considered. Exact
algorithm for simulation of measurements procedure with multiple radiation sources developed. Modelling results
are shown to have perfect agreement with calibration measurements.
PACS: 28.41Te
1. INTRODUCTION
Spherical detector (SD) device created in ISP NPP
was successfully used for gamma radiation angular
distribution studies for the “Shelter” object (SO) and
can be used for the similar purposes on other nuclear
power facilities. In our paper we consider mathematical
modelling of the gamma radiation angular distribution
measurement procedure for multiple radiation sources
with different intensities, which are placed at the
different distances.
2. MATHEMATICAL MODEL
The proposed method is based on the so-called
response function describing SD device detectors values
for point source with fixed position. In fact, they
constitute vector-function for the source angular
coordinates
)(),( det
32
det
1 H ,,Hii ≡ϕθ= iFiF . (1)
Calculation of the response functions provides the
way for attenuation coefficients corrention and
calibration results verification. Mathematical model of
the SD device required for calculations with GEANT-3
software was developed at NSC KIPT and numeric
computations were performed.
Response functions obtained by the mathematical
modeling can be used for the more precise method of
point source angular coordinates measurements and for
angular distribution procedure simulation in the case of
multiple radiation sources. Significant accuracy
improvement requires a considerably large set of
response functions for various radiation source
positions. Direct calculations for total spatial angle are
crucial enough due to large amount of numerical
calculations. We can essentially reduce number of
calculations taking into account the symmetry of
detection module.
Detection module construction implies detector
collimating holes placement the icosahedron’s vertices
(12 holes) and dodecahedron’s vertices (20 holes) with
entire symmetry corresponding to icosahedron’s spatial
symmetry group. It follows that applying symmetry
transformations to response function for one source
position we obtain response function for another source
location corresponding to the first location’s symmetry
transformation
FF R~=′ , (2)
where R~ is as symmetry transformation. Applying
symmetry transformation to the detector coordinates we
can find the rule for detector’s positions interchange
during detector module rotations.
So we can limit the set of response functions to
those belong to the single icosahedron’s face while the
others could be obtained using the symmetry
transformations (2) from this only set. During our
calculations this base face was 1-7-11 (the numbers
denote vertices defining the vertex according to
Table 1). In the center of this face collimating hole 2 is
located.
Table 1. Collimation holes angular coordinates
№ θ, deg. ϕ, deg. № θ, deg. ϕ, deg.
1 0 0 17 100.8 36
2 37.4 0 18 100.8 108
3 37.4 72 19 100.8 180
4 37.4 144 20 100.8 252
5 37.4 216 21 100.8 324
6 37.4 288 22 116.6 0
7 63.4 36 23 116.6 72
8 63.4 108 24 116.6 144
9 63.4 180 25 116.6 216
10 63.4 252 26 116.6 288
11 63.4 324 27 142.6 36
12 79.2 0 28 142.6 108
13 79.2 72 29 142.6 180
14 79.2 144 30 142.6 252
15 79.2 216 31 142.6 324
16 79.2 288 32 180 0
Symmetry transformation R corresponds to some
spatial rotation of the detector module. Such rotation [3]
is defined by three independent parameters. In our case
it is efficient to use 3×3 nonsingular matrix that is
coordinate system rotation matrix.
For the response functions calculations we need to
develop an algorithm to generate rotation matrix that
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2005, № 6.
Series: Nuclear Physics Investigations (45), p. 63-65. 63
mailto:khazhm@kipt.kharkov.ua
transforms icosahedron’s face 1-7-11 to any other face
preserving face’s orientation. We shall show this is
possible when both center and one of the vertices of the
new face are known. Table 2 contains numbers for
centers of planes together with corresponding vertices
numbers.
Table 2. Plane centers and corresponding vertices
numbers
Plane № Center number Vertex number
1 2 1, 7, 11
2 3 1, 7, 8
3 4 1, 8, 9
4 5 1, 9, 10
5 6 1, 11, 10
6 12 7, 11, 22
7 13 8, 7, 23
8 14 9, 8, 24
9 15 10, 9, 25
10 16 11, 10, 26
11 17 7, 23, 22
12 18 8, 24, 23
13 19 9, 25, 24
14 20 10, 26, 25
15 21 11, 22, 26
16 27 22, 26, 32
17 28 23, 22, 32
18 29 24, 23, 32
19 30 25, 24, 32
20 31 26, 25, 32
Firstly, let us fix the initial coordinate system. Initial
coordinate system has the center in the icosahedron’s
geometrical center with Z-axis directed to hole 1 (see
Table 1) and hole 2 belongs to XZ coordinate plane. Let
zn and 0n denote vector from the detector block’s
geometrical center to the vertex and to the center of new
plane correspondently. From these we can generate
three coordinate orts xe ′ , ye ′ and ze ′ for the rotated
coordinate system
.
,
,
0
0
zyx
z
z
y
z
z
z
eee
ne
nee
n
ne
×=′
×
×=′
=′
(2)
Thus we define the new coordinate system that has
Z-axis coinciding with zn and 0n vector belongs to XZ
plane. Rotation matrix R in this case could be
constructed from xe ′ , ye ′ and ze ′ orts
′′′
′′′
′′′
=
333
222
111
zyx
zyx
zyx
eee
eee
eee
R . (3)
Matrix R constructed this way defines
transformation from the initial coordinate system to new
coordinate system with vector transformation rule
,
3
1
∑
=
=′
k
kiki rRr (4)
where kr and ir ′ are components of the arbitrary vector
in initial and rotated coordinate systems.
According to the definition response function gives
detector counts for the point source for the known
position relative to detector module. This point source
could be thought as a model source. From the above it
follows that response function symmetry
transformations provide us with response functions for
other source’s positions being in turn the result of those
symmetry transformations. This fact could be used for
simulation of measurement procedure with multiple
radiation sources.
In the case of multiple sources we need a set of
precalculated response functions for the 1-7-11 plane.
Using symmetry transformations we can find
correspondence between the simulated source and one
of the response functions and the inverse transformation
gives us detector counts for the source. Total radiation
rate from multiple sources is additive so we can obtain it
by adding detector counts for each point source. We can
use weighted detectors counts to simulate sources with
varying intensity. The exact algorithm is presented below.
The following algorithm is applied for each source.
Starting parameters are source angular coordinates and
its relative intensity. Also we need to switch from
angular to Cartesian coordinates for detectors and
sources. Here we assume the corresponding points in
the Cartesian coordinate system to lie on the unit sphere.
Hence we can introduce unit vector s pointing to
source and a set of unit vectors { }miSS i
1, == ,
pointing to the locations that response functions were
calculated.
1. Among the detectors located in plane centers (see
Table 2) find the one closest to the given source. The
closeness criterion is distance minimum between
source and plane center. Cartesian coordinates of the
selected vertex give us vector zn
.
2. From the Table 2 choose vertex detectors for the
plane selected on the previous step and find the
closest to source. Cartesian coordinates of the
selected vertex give us vector 0n
.
3. Using (3) and (4) construct xe ′ , ye ′ and ze ′ orts and
transformation matrix R.
4. Apply transformation R to vector s according to (4)
to obtain rotated vector s′ , which belongs to 1-7-11
plane.
5. Using minimum distance criterion select from the set
S the closest vector .s′′ This one will be prototype
for the source.
6. Using the inverse transformation 1−R on vector s ′′
get the model source coordinates and the
corresponding detectors counts.
7. Multiply model detectors counts obtained on the
previous step by the model source relative intensity
and add to the resulting data.
64
The result of this algorithm applied to all sources it
total response function for the set of point sources with
known positions and relative intensities. For the actual
calculations this algorithm was implemented using C++
language.
Fig. 1. The result of modeling for two sources
Fig. 2. The result of calibration measurements for two sources
3. RESULTS AND CONCLUSIONS
For testing purposes we have performed calculations
for the same sources positions as those used for
calibration measurements (see Figs. 1,2). Perfect
agreement was achieved proving the correctness of
developed modeling method. Thus one can apply it for
more accurate SD device calibration procedure, detector
module angular resolution survey and for SD device
certification procedure.
REFERENCE
65
МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ ШАРОВОГО ДЕТЕКТОРА
С УЧЕТОМ СИММЕТРИИ ДЕТЕКТОРНОГО БЛОКА
В.Г. Батий, Д.В. Федорченко, И.М. Прохорец, С.И. Прохорец, М.А. Хажмурадов
Рассмотрено математическое моделирование шарового детектора с учетом свойств симметрии.
Представлен последовательный алгоритм моделирования процедуры измерения при наличии нескольких
источников излучения. Показано, что результаты моделирования имеют хорошее согласие с
калибровочными измерениями.
МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ ШАРОВОГО ДЕТЕКТОРА
З УРАХУВАННЯМ СИМЕТРІЇ ДЕТЕКТОРНОГО БЛОКУ
В.Г. Батій, Д.В. Федорченко, І.М. Прохорець, С.І. Прохорець, М.А. Хажмурадов
Розглянуто математичне моделювання шарового детектора з урахуванням властивостей симетрії.
Представлено послідовний алгоритм моделювання процедури вимірювань за наявності декількох джерел
випромінювання. Показано, що результати моделювання добре узгоджуються з калібрувальними
вимірюваннями.
66
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