Point-kernel method for radiation fields simulation
Point-kernel source method for radiation field calculation using Mercure-3 code is considered. Calculation results are shown to be in perfect agreement with those obtained using MCNP code.
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2007
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| Цитувати: | Point-kernel method for radiation fields simulation / I.M. Prokhorets, S.I. Prokhorets, M.A. Khazhmuradov, E.V. Rudychev, D.V. Fedorchenko // Вопросы атомной науки и техники. — 2007. — № 5. — С. 106-109. — Бібліогр.: 5 назв. — англ. |
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irk-123456789-1104002017-01-05T03:02:27Z Point-kernel method for radiation fields simulation Prokhorets, M.I. Prokhorets, S.I. Khazhmuradov, M.A. Rudychev, E.V. Fedorchenko, D.V. Ядернo-физические методы и обработка данных Point-kernel source method for radiation field calculation using Mercure-3 code is considered. Calculation results are shown to be in perfect agreement with those obtained using MCNP code. На прикладі програмного коду Mercure-3 розглянуто застосування метода точкового джерела для розрахунку радіаційних полів. Показано, що результати розрахунків добре узгоджуються з розрахунками за методом Монте-Карло, які виконано за допомогою програмного коду MCNP. На примере программного кода Mercure-3 рассмотрено применение метода точечного источника для расчета радиационных полей. Показано, что результаты расчетов хорошо согласуются с расчетами по методу Монте-Карло, выполненными при помощи программного кода MCNP. 2007 Article Point-kernel method for radiation fields simulation / I.M. Prokhorets, S.I. Prokhorets, M.A. Khazhmuradov, E.V. Rudychev, D.V. Fedorchenko // Вопросы атомной науки и техники. — 2007. — № 5. — С. 106-109. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 02.60.Cb, 28.41.Te, 28.52.Av http://dspace.nbuv.gov.ua/handle/123456789/110400 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Ядернo-физические методы и обработка данных Ядернo-физические методы и обработка данных |
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Ядернo-физические методы и обработка данных Ядернo-физические методы и обработка данных Prokhorets, M.I. Prokhorets, S.I. Khazhmuradov, M.A. Rudychev, E.V. Fedorchenko, D.V. Point-kernel method for radiation fields simulation Вопросы атомной науки и техники |
| description |
Point-kernel source method for radiation field calculation using Mercure-3 code is considered. Calculation results are shown to be in perfect agreement with those obtained using MCNP code. |
| format |
Article |
| author |
Prokhorets, M.I. Prokhorets, S.I. Khazhmuradov, M.A. Rudychev, E.V. Fedorchenko, D.V. |
| author_facet |
Prokhorets, M.I. Prokhorets, S.I. Khazhmuradov, M.A. Rudychev, E.V. Fedorchenko, D.V. |
| author_sort |
Prokhorets, M.I. |
| title |
Point-kernel method for radiation fields simulation |
| title_short |
Point-kernel method for radiation fields simulation |
| title_full |
Point-kernel method for radiation fields simulation |
| title_fullStr |
Point-kernel method for radiation fields simulation |
| title_full_unstemmed |
Point-kernel method for radiation fields simulation |
| title_sort |
point-kernel method for radiation fields simulation |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2007 |
| topic_facet |
Ядернo-физические методы и обработка данных |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/110400 |
| citation_txt |
Point-kernel method for radiation fields simulation / I.M. Prokhorets, S.I. Prokhorets, M.A. Khazhmuradov, E.V. Rudychev, D.V. Fedorchenko // Вопросы атомной науки и техники. — 2007. — № 5. — С. 106-109. — Бібліогр.: 5 назв. — англ. |
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Вопросы атомной науки и техники |
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2025-07-08T00:34:07Z |
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| fulltext |
POINT-KERNEL METHOD FOR RADIATION FIELDS
SIMULATION
I.M. Prokhorets, S.I. Prokhorets, M.A. Khazhmuradov, E.V. Rudychev,
D.V. Fedorchenko∗
National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
(Received January 6, 2007)
Point-kernel source method for radiation field calculation using Mercure-3 code is considered. Calculation results are
shown to be in perfect agreement with those obtained using MCNP code.
PACS: 02.60.Cb, 28.41.Te, 28.52.Av
1. INTRODUCTION
Design and maintenance of nuclear cycle facilities
mandatory imply implementation of efficient radia-
tion protection. Radiation environment assessment
and shielding optimization require a considerable
amount of numerical calculations. Nowadays math-
ematical modelling methods and corresponding soft-
ware are widely used for such kind of problems. All
the software used for radiation fields modelling falls
into two main groups. To the first belongs software
based on Monte Carlo methods, such as MCNP [1],
Geant [2], Penelope, Fluka, etc. Modern implementa-
tions of Monte Carlo method provide consistent ac-
counting of radiation transport effects. This leads
to perfect accuracy of the calculated values even for
complex models. At the same time Monte Carlo cal-
culations require considerable amount of computing
time. Computation burden increases rapidly for com-
plex geometries, multiple sources and thick shielding.
Another group contains software implementing
analytical methods. The examples of such programs
are Microshield, QAD, Mercure. All this programs
use point-kernel method for doze calculations. This
method is much more less computationally intensive
than Monte Carlo method. Series of calculations nec-
essary for shielding optimization could be conducted
at reliable time using point-kernel method. But due
to macroscopic approach to radiation transport this
methods lucks consistency. The main problem point-
kernel method encounters is account for scattered ra-
diation which is usually implemented through semi-
empirical approximation. Additional ”build-up” fac-
tor must be introduced as a multiplier to the atten-
uated doze. Determination of the appropriate build-
up factor can be rather complex as it depends upon
the energy, the thickness and type of material. Un-
certainties in determining build-up factor essentially
limit the accuracy of point-kernel method.
In our paper we considered some aspects of point-
kernel method and its implementation by Mercure-3
program code [4]. Results of doze calculations for
some typical cases are presented. Also point-kernel
method verification by the MCNP code [1] is dis-
cussed.
2. POINT-SOURCE KERNEL METHOD
Point-kernel method is macroscopic approach
used for gamma radiation exposure rate calculations.
Within this approach gamma radiation propagation
is assumed beam-like. Effects of radiation interaction
in matter are described using macroscopic linear at-
tenuation factors. Consistent scattered radiation ac-
counting could not achieved within macroscopic ap-
proach. So common practice is to use semi-empirical
relations, such as Berger formula, Taylor formula [3],
etc.
According to the main idea of point-kernel
method radiation source volume is cut up into ele-
mentary cells (point kernels). Each point kernel gives
contribution to the doze rate at the detecting point
for radiation energy E
A(~r, ~r′, E) = C(E)B(t, E)
exp(−t(E))
4π(~r − ~r′)2
, (1)
where C(E) is gamma flux density to doze rate con-
version factor, B(E) – build-up factor, t(E) – path
length along ~r−~r′ line measured in mean free pathes
(mfp)
t =
n∑
i=1
µi(E)Xi, (2)
where i is index of the space region, n – number of
regions, µi(E) – linear attenuation factor for i-th re-
gion and Xi – length of the section of ~r − ~r′ line
in i-th space region. Geometry used for point-kernel
calculations is shown on the Fig.1.
Relation (1) corresponds to beam-like radiation
propagation similar to that used in geometrical op-
tics. Factors µi describe radiation attenuation along
∗Corresponding author. E-mail address: d.fedorchenko@gmail.com
106 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2007, N5.
Series: Nuclear Physics Investigations (48), p.106-109.
the given path. From the microscopic theory it fol-
lows that attenuation coefficients µi depend on mat-
ter composition and on photon energy.
Build-up factor B(E) accounts for scattered radi-
ation. Mercure-3 implements Taylor formula
B(t) = β(E) exp(−α1(E)t(E))+
+ (1− β(E)) exp(−α2(E)t(E)), (3)
where α1, α2, β – parameters of Taylor for-
mula [3, 4]. This parameters depend on
material composition and on radiation energy
E. Mercure-3 code contains its own library
of build-up factors for commonly used materials.
Fig.1. Geometry of point-source kernel method
Point-source kernels are assumed to be indepen-
dent. So total doze rate at detection point is obtained
by integration of (1) over the source volume and sum-
mation over the energies Ek of radiation spectrum
A(~r) =
N∑
k=1
∫∫∫
V
d~r′A(~r, ~r′, Ek). (4)
For simple geometries integration in (4) could be
carried out analytically. A number of corresponding
relations could be found in handbook [3]. In general
case this integration is performed numerically.
Spatial integration in (4) presents a problem for
detection points located too close to source. This
is a common drawback of point-source kernel meth-
ods due to integral divergence at such points. One
of the possible solutions is to exclude from integra-
tion kernel points that fall within some predefined
region near the detection point. This procedure pro-
vides stabilization of integration procedure in (4) but
leads to some loss of accuracy. At the same time for
the most cases concerning radiation protection this
problem does not arise at all. The matter is that de-
tection points are usually separated from the source
by shielding layers and no divergence take place.
3. CALCULATION RESULTS AND
DISCUSSION
There exist several software implementations of
point-kernel method. Our choice of Mercure-3 code
was stipulated by several reasons. The main was the
highly verified code, as the program kernel has been
developed as early as in 1967 for three dimensional
gamma shielding calculations on IBM mainframes
and due to sufficient accuracy has been recommended
by EURATOM for general use [5]. Another valuable
feature of Mercure-3 code is rather powerful and flex-
ible geometry module capable of handling complex
structures with multiple sources or inhomogeneous
source. Also Mercure-3 contains gamma cross sec-
tion library for commonly used materials and a set of
Taylor coefficients of buildup factor (3). Addition of
new materials to the library is possible.
One of the nuclear cycle most important compo-
nents is radioactive waste management and storage.
Effective personnel radiation protection is necessary
during spent fuel processing. So we have considered
a problem of doze rate calculations for VVER-1000
nuclear reactor fuel assembly.
Technological operations with spent fuel assem-
bly are usually performed after cooling period.
Photon spectra of the spend fuel assembly de-
pends on cooling period duration. For our cal-
culations photon spectrum of spent fuel assembly
for 3 year cooling period was used (see table).
Fuel assembly photon spectra after 3 year cooling
period
Energy, MeV Activity, ph/s
0.015 3.27 · 1011
0.02 1.04 · 1011
0.03 1.96 · 1014
0.04 2.09 · 1014
0.05 9.27 · 1012
0.06 8.35 · 1012
0.08 3.73 · 1012
0.1 6.28 · 1013
0.15 1.60 · 1014
0.2 1.51 · 1013
0.3 1.05 · 1013
0.4 2.57 · 1013
0.5 3.33 · 1014
0.6 3.88 · 1015
0.8 1.43 · 1015
1.0 1.61 · 1014
1.5 1.59 · 1014
2.0 1.15 · 1013
Real fuel assembly is constituted by a number
of construction elements with 312 fuel elements in-
side. Detailed description of the fuel assembly inter-
nal structure is a rather complicated procedure. At
the same time the very common considerations show
details of internal structure do not influence signifi-
cantly on doze calculations. Thus for modelling pur-
poses we have neglected assembly internal structure.
Fuel assembly geometry was described by cylindrical
volumes - one for steel shank and another for bun of
fuel elements. Model geometry of fuel assembly used
for calculations is shown on the Fig.2. Bun of fuel ele-
107
ments was replaced by homogenous media composed
by uranium dioxide, zirconium and air. Volumetric
concentrations were chosen in accordance to assem-
bly technical specification: 0.3186(Zr), 0.205(UO2),
0.477(air).
Exposure doze rate calculations results are shown
on the Fig.3. Exponential doze rate attenuation with
distance is observed. Doze rate values are close to
those obtained earlier using Microshield code, which
is verified commercial product for such calculations.
Fig.2. Fuel source assembly geometry
Monte Carlo calculations for fuel assemblies and
similar objects are impeded by strong photon absorp-
tion in UO2 contained in fuel elements. Another im-
portant factor is large geometrical size of the VVER
fuel assembly. An intensive calculations are neces-
sary to achieve appropriate accuracy in this case.
Fig.3. Exposure doze rate for fuel assembly
In order to decrease computational burden re-
quired for Monte Carlo simulation a small VVER-M
assembly was taken as a source for verification calcu-
lations. Photon spectrum was the same as in previous
case (see table). Problem geometry included VVER-
M fuel assembly placed inside concrete shielding (see
Fig.4). Exposure doze rate was calculated for points
outside the shield to avoid computational instability
of point-kernel method.
Calculations for this geometry were carried out
both with Mercure-3 code and MCNP (Monte-
Carlo method). The results are shown on the
Fig.5. Perfect agreement between calculations us-
ing Mercure-3 and Monte-Carlo method (MCNP)
proves reliability of point-kernel method for radia-
tion fields modelling. At the same time computing
speed of Mercure-3 code is considerably higher than
those of MCNP (and other Monte-Carlo methods).
Fig.4. VVER-M assembly in concrete well
Fig.5. Exposure doze rate for VVER-M fuel
assembly
So we can consider point-kernel method imple-
mented by Mercure-3 code an efficient instrument
for various dose rate calculations, including shield-
ing optimizations. We have shown correct software
implementation of this method provides accurate re-
sults for practical needs. More over, optimization
scheme may include preliminary calculations using
point-kernel method with consequent Monte-Carlo
simulation.
This work was supported by STCU project 3511.
REFERENCES
1. J. Breismeister, Ed. MCNP - A General Monte-
Carlo N-Particle Transport Code. LA-13709-M.
Los Alamos National Laboratory: Los Alamos.
NM, 2000.
2. GEANT4 Physics Reference Manual. GEANT4
Working Group. CERN, June 21, 2004.
108
3. V.P. Mashkovich Protection for ionizing radia-
tion. Handbook., Moscow: ”Energoizdat”, 1982,
296p. (in Russian)
4. C. Devillers Programme MERCURE-3. Rapport
C. E. A. - R. 3264, Fontenay-aus-Roses, 1967.
5. ESIS Newsletter 5 (April 1973) 1. EURATOM
European Shielding Information Service, Ispra,
Italy.
МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ РАДИАЦИОННЫХ ПОЛЕЙ МЕТОДОМ
ТОЧЕЧНОГО ИСТОЧНИКА
И.М. Прохорец, С.И. Прохорец, М.А. Хажмурадов, Е.В. Рудычев, Д.В. Федорченко
На примере программного кода Mercure-3 рассмотрено применение метода точечного источника для
расчета радиационных полей. Показано, что результаты расчетов хорошо согласуются с расчетами по
методу Монте-Карло, выполненными при помощи программного кода MCNP.
МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ РАДIАЦIЙНИХ ПОЛIВ МЕТОДОМ
ТОЧКОВОГО ДЖЕРЕЛА
I.М. Прохорець, С.I. Прохорець, М.А. Хажмурадов, Є.В. Рудичев, Д.В. Федорченко
На прикладi програмного коду Mercure-3 розглянуто застосування метода точкового джерела для
розрахунку радiацiйних полiв. Показано, що результати розрахункiв добре узгоджуються з розрахун-
ками за методом Монте-Карло, якi виконано за допомогою програмного коду MCNP.
109
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