CC BY
9
УДК: 539.12
V.M. LAZURIK, V.T. LAZURIK, G. POPOV
V.N. Karasin Kharkiv National University
Z. ZIMEK
Institute of Nuclear Chemistry and Technology, Warsaw, Poland
ENERGY CHARACTERISTICS IN TWO-PARAMETRIC MODEL OF ELECTRON
BEAM
The relationships which are linking the parameters of the standard and the two-parametric models for the electron beam energy were obtained on the base of linear approximation of dependences for the spatial characteristics of the absorbed dose in aluminum as function of the electron beam energy. It was performed comparison of calculation results with measurements of electron beam energy by the dosimetric wedge method.
Keywords: electron beam energy, dosimetric wedge, two-parametric model, computational method
PFSEM.
В.М. ЛАЗУРИК, В.Т. ЛАЗУРИК, Г.Ф. ПОПОВ
Харювський нацюнальний ушверситет iM.B.H.Kapa3iHa
З.З1МЕК
1нститут Ядерно! Xirni i Технологш, Варшава, Польща
ХАРАКТЕРИСТИКИ ЕНЕРГП У ДВОПАРАМЕТРИЧНО1 МОДЕЛ1 ПУЧКА ЕЛЕКТРОН1В
На основi апроксимацИ залежностей просторових характеристик поглиненог дози в алюмтИ eid енергИ електротв в пучку, отриман спiввiдношення, як зв 'язують параметри стандартног i двопараметрично'1 моделi енергИ пучка електротв. Проведено порiвняння результатiв розрахунюв i вимiрювань енергИ електротв методом дозиметричного клину.
Ключовi слова: енергiя пучку електротв, дозиметричний клин, двопараметрична модель, розрахунковий метод PFSEM
В.М. ЛАЗУРИК, В.Т. ЛАЗУРИК, Г.Ф. ПОПОВ
Харьковский национальный университет им В.Н. Каразина
З.ЗИМЕК
Институт Ядерной Химии и Технологий, Варшава, Польша
ХАРАКТЕРИСТИКИ ЭНЕРГИИ В ДВУХПАРАМЕТРИЧЕСКОЙ МОДЕЛИ ПУЧКА ЭЛЕКТРОНОВ
На основе линейной аппроксимации зависимостей пространственных характеристик поглощенной дозы в алюминии от энергии электронов в пучке, получены соотношения связывающие параметры стандартной и двухпараметрической модели энергии пучка электронов. Проведено сравнение результатов расчета и измерений энергии электронов методом дозиметрического клина.
Ключевые слова: энергия пучка электронов, дозиметрический клин, двухпараметрическая модель, вычислительный метод PFSEM
INTRODUCTION
Computer simulation of process irradiation with electron beams (EB) of various materials can correctly to plan and control the performance of work on radiation processing installations [1]. In this case, some data characteristics of the irradiation process are required to perform modeling, in particular in the use of electron beams as source irradiation it is required the knowledge of EB energy characteristics. It is difficult to realize into practice the current direct control of the EB energy during products irradiation. Methods for determining the characteristics of EB energy with use of dosimetric devices such as dosimetric wedge and stack are described in the international standards, for example, [2,3] and in some scientific papers [4-6].
The special computational method for determination of EB energy characteristics on the base of two-parametric fitting of semi-empirical model for the depth-dose curves (DDC) of electrons beam to measurement results [7] (PFSEM method) is presented in the report [6]. The analysis of numerical experiments has shown that PFSEM method can effectively consider random displacements arising from the use of dosimetric wedge with a continuous strips of dosimetric films and minimizes the magnitude of the uncertainty value of the electron beam energy, calculated from the measurements.
In the paper [8] it is proposed to use two fitting parameters for PFSEM method as model parameters for simulation the electron beam irradiation process: E0 - the energy of mono-energetic and mono-directional electron source, X0 - the thickness of the aluminum plate, located in front of the irradiated object.
Approbation of this approach on working radiation-processing facility shows promising application PFSEM method as a means of obtaining baseline data on the characteristics of the electron beam, which are necessary for computer modeling of the irradiation process. In this paper we study the dependences, which are linking the characteristics (Ep -most probably energy, Eav -average energy) of the electron beam model, as described in the international standards [2,3], with the characteristics (E0, X0) two-parametric model [8], which provides simulation of irradiation processes with electron beams.
RELATIONS BETWEEN MODELS PARAMETERS OF ELECTRON BEAM
As example the Figure1 demonstrates the geometrical model of dosimetric wedge irradiated with non-diverging EB on moving conveyor. In the Fig. 1 two wedges are stacking together to form a rectangular block. Dosimetric film in form of strip is inserted along the sloping surface between the two wedges made of an arbitrary materials. Angle 9 should not be larger than 300 The rectangular block can be located under arbitrary angles relatively incident electron beam axis.
Fig. 1. Model of the dosimetric wedge with dosimetric film irradiated with scanned EB.
Axis X - direction of EB incidence, axis Y - direction of EB scanning, axis Z - direction of conveyer motion.
In the standard model the values of EB characteristics for the energy (Ep, Eav) are uniquely determined by the spatial characteristics (Rp, R50), which are obtained at processing the measuring results for the depth dependence of the absorbed dose (see such as [3]). Hereinafter in the text we will consider the distribution of the absorbed dose of electron beam along the length of dosimetric film which is located in an aluminum dosimetric wedge. The absorbed dose of electron beam along the length of dosimetric film can be easy recalculated into depth-dose distribution of the Al wedge (scale factor is 0.28 for the Al standard dosimetric wedge).
Figure 2 presents: the measurement results of the absorbed dose of electrons along length of the PVC dosimetric film (separate points, 3), corresponding to the experimental data which were obtained on the EB accelerator with EB energy 9.6 MeV at the INCT, Warsaw [8], and approximation of this data using PFSEM method (smooth curves, 1 and 2). PVC dosimetric films were located into standard RISO aluminum dosimetric wedge, see Fig.1.
35* \5
"l 20 / \ \ Model S i \\ V= -11.396; - 76.894 Experiment \\ R"= °-9996
y = -9.64SBx-S5.E8 \ \ R2 = 0.99S9 \\
Approximation . Tb\ y = -9.45S9X + ¡34.05B ' Vl
R: = 0.9597 I 'JA 6\
I I Wfc
-2-1 0 1 2 3 4 5 6 7 8 Length, cm
Fig. 2. The spatial parameters of models for the electron beam.
Solid lines (4 and 5) - linear interpolation for the back slope dose - illustrate the method of determining the value of Rp (corresponds to the intersection point of the axis length) to the experimental data and approximating curve.
The vertical dotted line marks 6 the position of the R50 - the length at which the dose is two times less than the maximum dose. The Figure 2 shows that fitting the semi-empirical model to the experimental data made with PFSEM method can correctly determine the spatial characteristics (Rp and R50) for the depth-dose curves (DDC) dependence of the absorbed dose of electron beam.
Interpretation of spatial parameter X0 in two-parametric model for EB is shown in Fig. 2. The vertical solid line (7) in the negative length indicates the entry point of the mono-energetic electron beam with an energy E0 in the aluminum plate with thickness X0, located in front of the Al dosimetric wedge. The vertical solid line (7) is located on the length before the zero mark.
On the basis of the physical interpretation of the parameters (E0, X0) for the electron beam model [8], the following relationships were obtained:
R*p (E0) = Rp + X0
R*50(E0) = R50 + X0
(1) (2)
where R *p(E) and R*50 (E) of electrons practical range and the depth at which the dose is less than twice the maximum value, depending on the electron energy E, which are defined in the semi-empirical model of electron radiation dose.
Calculations of the depth-dose curves in aluminum for various electron beam energies were performed to determine the dependency R*p(E) and R*50 (E) in the frames of semi-empirical model. Characteristics Rp and R50 were determined on the base of calculations results, in accordance with standard procedure [3]. Characteristics Rp and R50 are presented in the Table 1 and Fig.3.
y= 0.2092X- 0.0687 R2 = 1
£
o
CD
O
y = 0.1691x- 0.0965
R = 0.9997
0
8
10 12 14
Energy, MeV
Fig. 3. Approximation of dependencies for spatial characteristics Rp and R50 of the absorbed dose in aluminum as function of EB energy.
Linear approximation of the data allowed to get for R*p(E) and R*50 (E) the following expressions:
R*p(E) = 0.2092*E - 0.0687, R*50 (E) = 0.1691*E - 0.0965
(3)
As can be seen from the Fig.3, a linear function (the thin lines: 1 and 3) are well approximated the dependencies R*p(E) and R*50(E) in the electron energy range from 2 to 12 MeV. For comparison, the figure shows the results of calculations (the thick lines: 2 and 4), obtained by the formulas of the standard [3]:
E = 0.423 +4.69^Rp + 0.0523^Rp2
E = 0.734 +5.78^R50 + 0.0504^R5
(4)
(5)
As it is seen in Fig.3, the linear approximation (3) for dependencies R*p(E) and R*50(E) agrees well with the quadratic approximations (4), (5) dependences obtained in the prior physical investigations.
2
In accordance with presented in Fig.2 the equation for the linear interpolation of the experimental data, it is easy to calculate the value of Rp, and determine the value of Ep, in according to the equation (3). Depth dependence of the absorbed dose of monoenergetic electrons with energies Ep, is shown in Fig.2. As can be seen from the figure, the linear approximation (3) provides a good agreement between the spatial characteristics of Rp for experimental and the model depth-dose distributions.
The relation of spatial characteristics of the standard model - (Rp, R50) with parameters of the two-parametric model of the electron beam - (E0, X0) can be determined by considering the ratio of (1) and (2) as a system of two equations with two unknowns. At the same time, obtained linear approximations (3) for the dependencies R*p(E) and R*50(E) allow us to reduce the solution of the problem tasks to the solution of a system of two linear equations. Two-parametric model is linked to the spatial characteristics of the standard model of simple expressions:
E0 = (Rp - R50 - 0.0278)/0.04, X0 = 0.209E0 - 0.0687 - Rp (6)
Conversion of characteristics for the standard model out of parameters the two-parametric model is easy to perform with the following formulas:
Rp = 0.209E0 - 0.0687 - X0, R50 = 0.1691E0 - 0.0965 - X0 (7)
It should be note, that magnitude E0 for two-parametric model is uniquely determined only by the distance between two standard spatial characteristics for depth-dose distributions and therefore, regardless of the random shift arising from the use of a wedge with a continuous strip of dosimetric film.
COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS
The measurement results of the absorbed depth-dose distribution of EB within standard RISO dosimetric wedge were performed on the radiation facility with EB accelerator into INCT, Warsaw, Poland, with aim to test relations (6) and (7) [8].
The measurement results were processed by using of standard and PFSEM methods to determine parameters of the standard and two-parametric models of electron beams. Processing results are presented in the Table 1. The spatial parameters X0, Rp, R50 are shown in cm along length of the PVC dosimetric film. The results in the Table 1 which are marked with number 1, 2 and 3 were received in the PVC dosimetric films irradiated with EBs on radiation facility in Warsaw in routing experiments. The results in the Table 1 which are marked with Regime 1 and Regime 2 were received in the CTA dosimetric films irradiated with EBs in the special experiments [8].
The results calculations in accordance with equations (6) and (7) are presented in the Table 2. Calculation of parameters E0 and X0 in accordance with relations (6) was performed on the data base of parameters Rp and R50 from the Table 1. Spatial characteristics of Rp and R50 were calculated using equations (7) and Table 1. All calculated results in the Table 2 were obtained on the base of experimental data from Table 1.
Table 1. Experimental data related with parameters of the standard and two-parametric models of electron beams.
Experiment E0, MeV X0, cm Rp, cm R50, cm
1 9.9 0.764 6.46 4.85
2 9.47 0.112 6.64 5.25
3 11.21 1.188 6.96 5.2
Regime 1 10.01 1.1551 6.08 4.54
Regime 2 11.39 1.51 6.74 5.05
Table 2. Results calculations of parameters for the standard and two-parametric model of the electron beams.
Experiment E0, MeV X0, cm Rp, cm R50, cm
1 10.56 1.19 6.39 4.87
2 9.02 -0.14 6.72 5.26
3 11.62 1.47 6.94 5.24
Regime 1 10.08 1.20 6.08 4.55
Regime 2 11.12 1.33 6.76 5.02
As it is seen from comparison of the data in Tables 2 and 3, the spatial characteristics of Rp, R50, into standard model can be determined with good accuracy (<3%) for the parameter values E0, and X0 in two-parametric model. Calculation of parameters E0, and X0 for two-parametric model with use the spatial characteristics of Rp, and R50 of the standard model will give a large relative errors.
To assess the impact of errors in two-parametric model parameters on the errors of approximation of the depth-dose distribution, the Figure 4 provides the following information:
• experimental data (individual characters),
• approximation using two-parameter model of the electron beam with the parameters obtained PFSEM method (solid lines), and
• the parameters calculated in according to relationships (6) (dashed curves).
As can be seen from a comparison of curves, even considerable relatively errors in the parameters of two-parametric model E0 and X0 do not lead to drastic changes in the approximating curve and thus can be used in the practice of the radiation-technological centers.
0 1 2 3 4 5 6 7
Lengtb, cm
Fig 4. The approximation of the experimental data with the use of a two-parametric model of the electron beam.
RESULTS AND CONCLUSIONS
Obtained the systematic set of data and performed the linear approximation for the practical range of electrons R*p(E) and the depth of the half-dose reduction R*50(E) in the aluminum target, depending on the energy of electrons E. For the linear approximation R*p(E) and R*50(E) - obtained the relations linking the set of model parameters. It is shown that semi-empirical model of depth-dose distribution of electron radiation can be used to calculate the standard characteristics of the electron beam energy. Approximation curves obtained PFSEM method can be used instead of experimental data at determining the values (Rp and R50).
REFERENCES
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