Formulation of criterion functional and set of constraints in problems of physical settings designing
Characteristics of semiconductor spectrometer and neutronography setting were investigated using system analysis methods.
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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| Цитувати: | Formulation of criterion functional and set of constraints in problems of physical settings designing / I.M. Prokhorets, S.I. Prokhorets, M.A. Khazhmuradov // Вопросы атомной науки и техники. — 2004. — № 5. — С. 108-111. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-805492015-04-19T03:02:06Z Formulation of criterion functional and set of constraints in problems of physical settings designing Prokhorets, I.M. Prokhorets, S.I. Khazhmuradov, M.A. Применение ядерных методов Characteristics of semiconductor spectrometer and neutronography setting were investigated using system analysis methods. Розглянуто вплив параметрів напівпровідникового спектрометру з CdZnTe на енергетичну роздільність та вплив параметрів системи формування пучку на потік нейтронів для нейтронографічного пристрою. Рассмотрено влияние параметров полупроводникового спектрометра на основе CdZnTe на энергетическое разрешение и влияние параметров системы формирования пучка на поток нейтронов для нейтронографической установки. 2004 Article Formulation of criterion functional and set of constraints in problems of physical settings designing / I.M. Prokhorets, S.I. Prokhorets, M.A. Khazhmuradov // Вопросы атомной науки и техники. — 2004. — № 5. — С. 108-111. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 61.20ja http://dspace.nbuv.gov.ua/handle/123456789/80549 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Применение ядерных методов Применение ядерных методов |
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Применение ядерных методов Применение ядерных методов Prokhorets, I.M. Prokhorets, S.I. Khazhmuradov, M.A. Formulation of criterion functional and set of constraints in problems of physical settings designing Вопросы атомной науки и техники |
| description |
Characteristics of semiconductor spectrometer and neutronography setting were investigated using system analysis methods. |
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Article |
| author |
Prokhorets, I.M. Prokhorets, S.I. Khazhmuradov, M.A. |
| author_facet |
Prokhorets, I.M. Prokhorets, S.I. Khazhmuradov, M.A. |
| author_sort |
Prokhorets, I.M. |
| title |
Formulation of criterion functional and set of constraints in problems of physical settings designing |
| title_short |
Formulation of criterion functional and set of constraints in problems of physical settings designing |
| title_full |
Formulation of criterion functional and set of constraints in problems of physical settings designing |
| title_fullStr |
Formulation of criterion functional and set of constraints in problems of physical settings designing |
| title_full_unstemmed |
Formulation of criterion functional and set of constraints in problems of physical settings designing |
| title_sort |
formulation of criterion functional and set of constraints in problems of physical settings designing |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2004 |
| topic_facet |
Применение ядерных методов |
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http://dspace.nbuv.gov.ua/handle/123456789/80549 |
| citation_txt |
Formulation of criterion functional and set of constraints in problems of physical settings designing / I.M. Prokhorets, S.I. Prokhorets, M.A. Khazhmuradov // Вопросы атомной науки и техники. — 2004. — № 5. — С. 108-111. — Бібліогр.: 7 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT prokhoretsim formulationofcriterionfunctionalandsetofconstraintsinproblemsofphysicalsettingsdesigning AT prokhoretssi formulationofcriterionfunctionalandsetofconstraintsinproblemsofphysicalsettingsdesigning AT khazhmuradovma formulationofcriterionfunctionalandsetofconstraintsinproblemsofphysicalsettingsdesigning |
| first_indexed |
2025-07-06T04:33:42Z |
| last_indexed |
2025-07-06T04:33:42Z |
| _version_ |
1836870714610679808 |
| fulltext |
FORMULATION OF CRITERION FUNCTIONAL AND SET OF
CONSTRAINTS IN PROBLEMS OF PHYSICAL SETTINGS DESIGNING
I.M. Prokhorets, S.I. Prokhorets, M.A. Khazhmuradov
National Science Center “Kharkov Institute of Physics and Technology” KIPT, Kharkov, Ukraine
e-mail: khazhm@kipt.kharkov.ua, iprokhorets@kipt.kharkov.ua
Characteristics of semiconductor spectrometer and neutronography setting were investigated using system analy-
sis methods.
PACS: 61.20ja
1. INTRODUCTION
It is impossible to develop at NSC KIPT researches
in different branches of physics, technics and medicine
without creation new experimental settings at existing
and planned accelerators. At development new settings
it is necessary to choose from the set of feasible solu-
tions the best or optimal one. Problem of obtaining of
optimal solution exists at all development stages. Math-
ematically, search for optimal solution of such tasks re-
duces to finding parameters that give maximal or mini-
mal value of criterion functional [1].
There are different types of physical settings and
hence, there are different methods of their optimization.
But in spite of this there are stages common to develop-
ing of all physical settings. They are:
problem statement;
creation of physical model of design object;
optimization problem definition;
creation of mathematical model that depicts inter-
relationships between object main features;
problem solving on the basis of the used mathe-
matical model;
obtained results analyses, correction of conceptual
and mathematical models.
Data about object purposes and its operating regimes
are source information for optimal parameters finding.
They determine main design aim and requirements to
designed object. Influence of many factors on the design
object can be found using mathematical optimization
methods, which are subject of operation research or,
widely, of systems analysis [2].
System analysis methods are used in different
branches: military science, economics, agriculture,
medicine, etc. In spite of qualitative difference tasks in
all these branches of human activity reduce to choosing
of modus operandi and design parameters, that is, to de-
cision-making. It is concerned also such complex ob-
jects as technological and physical settings, where radia-
tion technologies are used. Till recently system analyses
of their characteristics wasn’t made, and used for this
purposes figure of merit characterize in most cases only
one feature. So it is actual to develop methods of physi-
cal settings and their systems characteristics defining at
the design stage using computer experiment methods.
The aim of this article is to investigate characteris-
tics of specific physical settings and their systems using
system analysis methods.
2. PHYSICAL MODEL
OF EXPERIMENTAL SETTING
Generally nearly each physical setting can be repre-
sented as set of subsystems with interrelationships
(Fig. 1), caused by system functional features [3,4].
Semiconductor spectrometer is one of the simplest sys-
tems. It consists from detector, prime- and shaping am-
plifier, bias voltage supply unit, converter of signal am-
plitude or charge to digital value. More complex system
is modern multichannel semiconductor detector- pixel,
drift or strip detector. Last one consists from many ex-
panded p-n junctions, each of them is separate detector
element with prime- and shaping amplifiers. Signal
from each amplifier is put to memory, read out, convert-
ed to digital value and put to intermediate or PC memo-
ry. Distinctive feature of such complex detector system
allocated at single semiconductor plane is relations be-
tween them.
1
2
А
11
А
22 3
E e
I e
АА
А
11
22 3 22 3
1111
22 3
АА2А1
Fig. 1. Simplified structure of physical setting. А1 …
А7 – physical setting subsystems. А0 – environment
Experimental setting can consist from several de-
vices based on different physical principles. Then it can
be represented as set of subsystems, each of them con-
sists from unit etc. On Fig. 1 environment influence
(temperature, moisture, irradiation) also is represented
as several unit А0. If each system doesn’t depend on the
next one, then optimization problem of its characteris-
tics reduces to autonomous modeling of subsystems.
Modeling can begin from any subsystem (unit), which
receive information from the single source. Representa-
108 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2004, № 5.
Series: Nuclear Physics Investigations (44), p. 108-111.
tion of complex system as set of subsystems allows to
use both mathematical models and information in form
of tables, diagrams etc. for characteristics of its subsys-
tems.
3. MATHEMATICAL DEFINITION
OF OPTIMIZATION PROBLEM
Each physical device has set of features that deter-
mine its purpose and can be changed or calculated.
Physical device features are called characteristics.
Characteristics can be independent each from another,
but they depend on directed parameters and external
factors which forms environment where device works.
Directed parameters and external factors are indepen-
dent variables and characteristics depend on them.
Parametric optimization deals with calculation of pa-
rameters ),...,,( 21 mxxx=X . It results in values ix , at
which criterion functional ( )XQ is maximum or mini-
mum. As criterion functional we can use, for example,
spatial, time, amplitude resolution or efficiency of radia-
tion registration. Let’s criterion functional has to be
minimal
)(min X
X
Q
gX∈ , (1)
where gX – region of allowed parameters. Such prob-
lem must be solved when following inequation system
is satisfied
njxxx jjj ,,2,1, =≤≤ +− , (2)
( ) miiii ,,2,1, =ϕ≤ϕ≤ϕ +− X , (3)
where −
jx , +
jx – values of j-th directed value charac-
terizing its allowed values range, and −ϕ i , +ϕ i – limit
value of characteristics.
Optimization problem (1-3) is solved using linear
(non-linear)-programming technique if optimality crite-
rion – criterion functional and constraints are linear
(non-linear) functions of parameters. If there are no
non-linear constraints (3), then solving of optimization
problem (1) reduces to minimum search of the criterion
functional (1) with constraints (2) that simplifies the
problem. Depending on number of variables optimiza-
tion problems can be one-dimensional (n = 1) or multi-
dimensional (multiparametric) (n ≥ 2).
When during design there is a need to obtain the best
values for several object characteristics it is necessary to
find such values of directed parameters, which give
minimum of criterion functional that satisfies all criteri-
ons simultaneously. It is necessary to find compromise
solving. Mathematically let’s introduce vector criterion
of optimality [5]
( ) ( ) ( ) ( )( )XXXQXQ sQQQ ,...,, 21= . (4)
Compromise solving of such multicriterion problem
is point X*∈Xg, which satisfies inequation Q(X*)≤Q(X).
Practically point *X search reduces to search for set
of partial optimization criterions, satisfying Pareto crite-
rion [2,5]. This criterion says that none of the partial cri-
terions can be diminished without increasing of the oth-
ers.
Let’s consider methods of determining of optimal
parameters of spectrometric channel with planar semi-
conductor detector and beam formation system for neu-
tronography setting. Direct search method can be used
to solve such optimization problem. Used for these pur-
poses algorithms allow to solve such tasks in a follow-
ing sequence: next variant generating, variant rating of
merit and decision-making.
4. OPTIMIZATION OF PLANAR SEMICON-
DUCTOR DETECTOR CHARACTERISTICS
Planar semiconductor detectors are widely used for
registration and spectrometry of radiation of different
types. Noise level is one of the characteristics, determin-
ing quality of channel for information read-out from
such device. If we consider that planar detector from
wide-gap semiconductor (CdTe, CdZnTe, GaAs) be-
haves as ionization chamber, then dark current [6]
ρd
VAId = , (5)
where V – potential difference applied to the parallel
contacts of the detector, d, A – detector thickness and
area and ρ – resistivity. Dark current defines parallel
noise in detector – charge sensitive preamplifier system
q
IENC d
p
τ= , (6)
where τ – integration time of the amplifier. Second
noise component – series noise
( )
τ
++=
m
stdgs g
kTCCC
q
ENC 41
, (7)
where dC – planar detector capacitance, gC – capaci-
tance of the gate of the input FET, stС – stray capaci-
tance associated with the connection of the amplifier,
mg – transconductance of the readout FET, k – Boltz-
mann constant, T – temperature. If two above-men-
tioned noise components are considered to be statistical-
ly independent then resulting electron noise
222
sp ENCENCENC += . (8)
So, if we consider electronic noise value as spec-
trometer characteristics, then mathematically minimiz-
ing of the noise can be considered as criterion functional
of such system. Criterion functional depends in our ex-
ample mainly on detector thickness and area, electric
field intensity in the detector volume, capacitance of the
gate of the input FET and FET parameters, integration
time of the amplifier and external factors – temperature,
irradiation, humidity. Integration time τ depends, in
turn, on drift time to outer contacts of the charges –
electrons and holes – born in the detector, i.e. on elec-
tron and hole mobility, detector thickness and electric
field intensity in the detector volume. Hence, criterion
functional depends on the following directed parameters:
– semiconductor detector thickness, mm – ≤ 10;
– semiconductor crystal area, cm2 – ≤ 100;
109
– electric field intensity, kV/cm – ≤ 2;
– capacitance of the gate of the input FET, pF – ≤ 2,5;
– transconductance of the readout FET, mS – ≥ 4;
– stray capacitance associated with the connection of
the amplifier, pF – ≤ 10;
– integration time of the amplifier, μs – ≤ 20.
On Fig. 2 it is shown results of spectrometer model-
ing with CdZnTe detectors with thickness 10 mm and
volume 1 and 10 cm3. Calculations was made for
V = 1 kV, gC = 2,5 pF, sC = 10 pF and mg = 6 mS.
Noise in electrons (rms) was converted to keV (FWHM)
using formula
ENCFWHM ⋅ξ⋅= 35,2 , (9)
where ξ = 5 eV – mean energy required to create an
electron-hole pair in CdZnTe.
t m, s
FW
H
M
, k
eV
1
10
10-1 1
100
10 cm 3
1 cm3
10
С = 10 pFs
FW
H
M
, k
eV
1
10
1
100
10-1
τ µ, s
10
10 cm 3
1 cm3
С = 0 pFs
Fig. 2. Noise (FWHM) versus integration time τ for
spectrometric channel with CdZnTe detectors with vol-
umes 1 and 10 cm3
From Fig. 2 it is clear that electronic noise allows to
obtain resolution near 7 keV (FWHM) with detector
1 cm and 1 cm2, that gives ∼1 % at the source 137Сs
(662 keV). Modern technologies of detector production
don’t allow obtain such resolution, as there are addition-
al noise sources, which increase FWHM. Minimum
points of the curve at Fig. 2 are solving of optimization
problem
minmin, →τ→
∈
FWHM
gXX (10)
Direct search method was used for obtaining this
minimum. In the minimum point we obtain compromise
between amplitude resolution and system operating
speed.
5. DETERMINING OF CHARACTERISTICS
OF NEUTRONOGRAPHY SETTING
Neutron flux density in full energy range and in sep-
arate energy intervals (thermal, fast and so on) are the
main characteristics of setting for neutron radiography
(NR). If NR setting is planned at the electron accelerator
base theses characteristics depend on such directed pa-
rameters as accelerated electrons energy and current at
neutron-producing target, target thickness and material,
collimator-moderator material and geometrical sizes
(Fig. 3).
e
T
H l L
Cd
Ф ОИ D
d
K
-
Fig. 3. Neutron beam shaping system (in a simplified way)
As criterion functional for neutronography setting
we can consider maximum number of neutrons with
specified energy spectrum when ratio of the hollow col-
limator length to its outer diameter is given. For this aim
we calculated neutron flux and energy spectrum after
collimator which we considered as cylinder with outer
diameter 40 cm and inner channel in the form of hollow
cylinder with diameter d = 10 cm. In our calculations
we used method of statistical testing [7]. During our cal-
culations we determined how moderator front wall
thickness, inner hollow cylinder length, distance from
object of researches to collimator out, neutron-produc-
ing target, cadmium inset influence on the beam charac-
teristics. We considered two targets: lead ball with di-
ameter 60 mm, placed at collimator axes and lead plate
with thickness 6 cm and diameter 10 cm, placed at angle
45° to electron beam and collimator axis. Both ball and
plate were isotropic neutron sources.
On Fig. 4 it is shown influence of hollow cylinder
length on the thermal neutron flux at the output plane of
the collimator-shaper. Modeling results show that neu-
tron–producing target in form of plane disk scanned
with electron beam gives in considered geometry larger
neutron flux, compared with ball and point beam. If we
consider that at energy 23 MeV neutron yield at 4π an-
gle is 4⋅1010 neutron/μA⋅s, such target allows to obtain
240 neutron/s of thermal neutrons with energies 0,025…
0,1 eV for d/L = 0,025. Influence of cadmium insert
(cylinder with wall thickness 1 mm and length 50 cm in-
side hollow cylinder of collimator) is shown on Fig. 5. It
is clear that influence of cadmium on the neutron flux is
insignificant when d/L = 0,025.
110
10
-8
10
-7
10
-6
10
-5
0 50 100 150 200 250 300 350 400
flu
x,
1
/c
m
2
L, cm
a)
10
-8
10
-7
50 100 150 200 250 300 350 400
flu
x,
1
/c
m
2
L, cm
b)
Fig. 4. Neutron flux from neutron-producing target in form of ball (a) and plane disk with diameter 10 cm (b) ver-
sus length of collimator-shaper hollow cylinder
10
-8
10
-7
50 100 150 200 250 300 350 400
flu
x,
1
/c
m
2
L, cm
flu
x,
1
/c
m
2
flu
x,
1
/c
m
2
Fig. 5. Influence of cadmium insert on the neutron
flux at the output of the collimator-shaper hollow cylin-
der. • – without cadmium insert; ▲ – with cadmium
6. CONCLUSIONS
In this work it is shown that problems of the physi-
cal settings development can be formulated as optimiza-
tion problems of complex systems. It was formulated
criterion functional and defined main parameters in the
set of constraints. It was solved several practical tasks,
appearing while developing semiconductor spectrome-
ters and settings for neutron researches.
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ФОРМУЛИРОВКА ФУНКЦИИ ЦЕЛИ И СИСТЕМ ОГРАНИЧЕНИЙ
В ЗАДАЧАХ ПРОЕКТИРОВАНИЯ ФИЗИЧЕСКИХ УСТАНОВОК
И.М. Прохорец, С.И. Прохорец, М.А. Хажмурадов
Рассмотрено влияние параметров полупроводникового спектрометра на основе CdZnTe на энергетиче-
ское разрешение и влияние параметров системы формирования пучка на поток нейтронов для нейтроногра-
фической установки.
ФОРМУЛЮВАННЯ ФУНКЦІЇ ЦІЛІ ТА СИСТЕМ ОБМЕЖЕНЬ
В ЗАДАЧАХ ПРОЕКТУВАННЯ ФІЗИЧНИХ УСТАНОВОК
І.М. Прохорець, С.І. Прохорець, М.А. Хажмурадов
Розглянуто вплив параметрів напівпровідникового спектрометру з CdZnTe на енергетичну роздільність
та вплив параметрів системи формування пучку на потік нейтронів для нейтронографічного пристрою.
I.M. Prokhorets, S.I. Prokhorets, M.A. Khazhmuradov
National Science Center “Kharkov Institute of Physics and Technology” KIPT, Kharkov, Ukraine
OF EXPERIMENTAL SETTING
3. MATHEMATICAL DEFINITION
OF OPTIMIZATION PROBLEM
5. DETERMINING OF CHARACTERISTICS OF NEUTRONOGRAPHY SETTING
6. CONCLUSIONS
І.М. Прохорець, С.І. Прохорець, М.А. Хажмурадов
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