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Simple method of evaluating absorbed dose in electron-beam processing
0146-5724/89 $3.00+0.00 Copyright © 1989PergamonPress plc
Radiat. Phys. Chem. Vol. 33, No. 5, pp. 411~t16, 1989 Int. J. Radiat. Appl. Instrum. Part C
Printed in Great Britain. All rights reserved
SIMPLE METHOD OF EVALUATING ABSORBED DOSE IN ELECTRON-BEAM PROCESSINGt TATSUO TABATA,1 RINSUKEITO,1 ISAMU KURIYAMA2 and YASUYUKIMORIUCHI3 'Radiation Center of Osaka Prefecture, Sakai, Osaka 593, 2Takasaki Branch, Irradiation Development Association, Takasaki, Gunma 370-12 and 3Gifu College of Medical Technology, Seki, Gifu 501-32, Japan Abstract--The use of semiempirical multilayer depth-dose code EDMULT for evaluating the absorbeddose coefficient K was studied with a personal computer. When multiplied by the charge per unit area, the coefficient gives the adsorbed dose in the sample irradiated by electrons. This relation presupposes the model configuration of the plane-parallel beam normally incident on the four-layer slab absorber. The absorber consists of the accelerator window, the air layer, the sample and the substratum. The initial energies of electrons above 100 keV were considered. The values of K obtained agreed reasonbly well with the values in the literature. The absorbed doses estimated from K were compared with the values measured by the radiochromic dosimeter for various sets of irradiation conditions. The r.m.s, deviation of the former from the latter was 6.7%. A calculated example of the effects of different substratum materials is given.
l. INTRODUCTION In planning the radiation processing with the lowenergy electron beam, it is important to estimate the absorbed dose in the sample. Electrons suffer appreciable energy-loss and scattering in the accelerator window and an air layer before entering the sample. Under the sample, there is often a substratum consisting of a metal, which has a considerable backscattering effect. Therefore, we have to treat the transport problem of electrons through a multilayer absorber with three or four components. MonteCarlo code systems to solve such problems are available for general use; examples are Integrated T I G E R Series (ITS) codes (Halbleib and Mellhorn, 1984). Recently Entinzon (Entinzon, 1986) has developed a method to solve the energy-transport equation numerically, and has evaluated dose distributions in multilayer absorbers. Both the Monte-Carlo codes and the numerical method, however, require a considerable time on a large computer. Therefore, it is desirable to develop a simple and accurate method for estimating the absorbed dose. Okabe et al. (Okabe et al., 1974) have proposed a simple relation to evaluate the absorbed dose for the standard geometry of electron irradiation. Their relation gives the absorbed dose as the product of the charge per unit area and a coefficient K. They have estimated the value of K for 300-keV electrons. Tabata (1975) has also given values of K for the fThis work was performed as an activity of Low-Energy Electron-Dose Measurement Committee sponsored by Irradiation Development Association. A preliminary account of part of this work was given at the Conference on Radiation Curing Asia, Tokyo, October 1986.
energies between 300 keV and 2 MeV. Proksch et al. (1979) have given a value of a coefficient, which can be reduced to K, at 500 keV. Tanaka et al. (1989) have proposed an approximate relation for K applicable to conditions of practical irradiation with lowenergy electrons. In this work, we studied a method to rapidly evaluate K for the sample in a multilayer of three or four components. A semiempirical three-layer depth-dose code EDMULT developed by Tabata and Ito (1981a, b, 1982, 1983) was used on a personal computer. Accuracy was checked by comparing (i) values of K with those reported by previous authors (Okabe et al., 1974; Tabata, 1975; Proksch et al., 1979) and (ii) the absorbed doses estimated from K, with measured values for various sets of irradiation conditions. 2. FORMULATION OF THE PROBLEM
We consider a model configuration for the standard irradiation on a conveyer with the low-energy electron accelrator. The configuration, originally used by Okabe et al. (1974) is as follows: the planeparallel beam is normally incident on a four-layer slab absorber. The first layer is the accelerator window made of aluminium or titanium foil of thickness x,.. The second is the air layer of thickness x a. The third is the sample layer of thickness x,. The fourth is the substratum of enough thickness to stop all the incident electrons (effectively semiinfinite thickness). This configuration is equivalent to the actual one for the accelerators of linear-filament type and broadbeam type. It is also a good approximation for the accelerator of scanning-beam type when the maximum angle of scanning is small.
411
412
TATSUOTABATAet al.
Our problem is to evaluate the absorbed dose D averaged over the thickness x~ of the sample layer. We treat the two cases of substratum materials (i) the same as and (ii) different from the sample material. The first case occurs, for example, when the absorbed dose in the uppermost sheets in a thick stack of film is considered. We call case (i) the three-component problem, and case (ii) the four-component problem. The following quantities are usually known: i0 = beam current in the accelerator tube (in A); v = conveyer speed (in m/s); w = beam width perpendicular to the movement of the conveyer at the sample surface (in m); E 0 -- initial energy of electrons when incident on the window (known from the accelerating voltage). From the first three quantities, we define effective charge-density q (charge per unit area) by: q = io/vw (in C/m2).
(1)
The product vw corresponds to the area of the irradiation field divided by the irradiation time. While we treat a multilayer including the window, we used here this product evaluated at the sample surface. The reason is as follows. In the depth-dose code EDMULT we used, the irradiation field assumed is of infinite area. So this code does not include the broadening of the field with increasing depth caused by scattering of electrons in the medium. To use the code for a case in which the broadening effect is appreciable, therefore, we must take it into account somehow. We can do this by using the field area at the depth in the absorber where dose is to be evaluated. This is equal to the field area at the sample surface (the linear thickness of the sample is usually negligible compared with that of the air layer). Using q, we can express the absorbed dose D by (Okabe et al., 1974): D = Kq
(in kGy)
(2)
Here K is defined by I i ~ + x a + x~
K = (102/x~)
l(x, Eo) dx ~¢+ Xa
(in kGy m2/C).
(3)
We call it absorbed-dose coefficient. In equation (3), I(x, Eo) is the energy deposition (in MeV cm2/g electron) by the electrons of initial energy E0 at the depth x in the multilayer absorber. The factor 102 stands for the arrangement of the units. Aside from this factor, K is equal to the absorbed energy per unit thickness per incident electron, averaged over the sample thickness. In the present model, the problem of evaluating the absorbed dose is thus divided into the problems of evaluating q and K. The present work treats the evaluation of K. As for the evaluation of q, the following must be noted. In some accelerators, the
pipe for coolant attached to the outside of the window intercepts a fraction of the beam. Then we must use as i0 the beam current in the accelerator tube reduced by the intercepted fraction. It is also necessary to take into account the gradient of the current density at the edges of the irradiation field in evaluating w. This can be done by using the "half-width" of the current-density distribution. 3. C O M P U T A T I O N A L
METHOD
3.1. The code used
We used the code EDMULT to compute the absorbed-dose coefficient K. This code was developed to evaluate depth~lose distributions produced by plane-parallel electron beams normally incident on multilayer slab absorbers (Tabata and Ito, 1981a,b, 1982, 1983). The numbers of layers permitted are from one to three. The last layer in the three-layer absorber must be effectively semiinfinite. The information obtainable are differential and integral energy-depositions as a function of depth in the absorber. The code is applicable to the initial energies from 0.1 to 20 MeV and to almost all materials including mixtures and compounds. The algorithm of the code uses semiemipirical realtions, and most of the calculations are made analytically. In the course of the present work, minor errors were found in the original code, and were removed. This improved agreement with experimental and Monte-Carlo results (cited in Tabata and Ito, 1981a) at the smallest depths in the third layer. Doses at these depths are most important in evaluating K for the three-component problems. The code EDMULT, originally written for the large computer, was adapted to a personal computer. An application program of EDMULT was developed to compute K and depth-dose distributions. The majority of subprograms are coded in F O R T R A N 77. The program can run on the operating system MS-DOS version 3.10 without modification for NEC PC-9800 series machines. Use of the machine with or without a numerical coprocessor is permitted. When the machine is equipped with a numerical coprocessor, the computation time required is quite small (a few seconds for a value of K). A floppy-disk copy of this program is available on request. 3.2. The three-component problem
The configuration of the three-component problem is the same as that assumed in EDMULT. In this problem, therefore, application of E D M U L T to the computation of K is simple. To evaluate the integral in equation (3), we call EDMULT twice for the depths corresponding to the limits of the integral. For the atomic number Z and the atomic weight A of air, the effective values given by the following formulas are used: Z = ~ f Z,; i
(4)
Absorbed dose and electron-beam processing
413
Table 1. Conditions assumed in evaluating the absorbed-dose coefficient K shown in Fig. I. The symbols x,., x~ and x, represent the thickness of the window foil, the air layer and the sample layer, respectively. For x~= 0, the value of K gives the surface dose when used in equation (2). The substratum material assumed is the same as the sample material except Okabe et al. (1974) who assumed aluminium x.. xa Sample x~ Author (,urn) (cm) material (#m) Present work (1) Present work (2) Okabe et al. (1974) Tabata (1975) Proksch et al. (1979)b
30 30 30 30 25
8 8 8 8 22
PETI~ PETP PETP PETP Nylon
0 100 100 100 0
~Polyethylene terephthalate, bAngle of incidence was assumed to be 15'>.
4. EXPERIMENTAL METHOD
Here f~ is the fraction by weight of the constituent element with atomic number Z~ and atomic weight A i. We use these equations also for the sample material consisting of a c o m p o u n d or a mixture. A different definition of the effective atomic number is described in the Appendix. 3.3. The f o u r - c o m p o n e n t problem
To treat the four-component problem by EDM U L T , we consider to reduce it to the threecomponent problem. We do this by treating the window and the air layer as a single composite-layer. The effective atomic number and atomic weight for the composite layer can be calculated again from equations (4) and (5). In this use of these equations, the fraction by weight f is equal to the mass-thickness fraction. To check the validity of this treatment, we compared the values of K for the three-component problems with those obtained by the regular treatment. The result of the comparison is described in Section 5.3. k
6ec
'
I , ,,r
>.
l',/""\ '/x ~ I \ \L
,oo
I
F
×
OKABE
•
TAB,
o .....
o
'
I
' ''
ET AL.
,,
PROKSCN ET , L
EO.
Ce~
"' PRESENT
20C
o m 0.1
0.5 INITIAL
I ENERGY
5
I0
(MeV1
Fig. 1. Absorbed-dose coefficient as a function of initial energy. The conditions assumed in the previous and present evaluations are given in Table 1. Dashed line: equation (6) for polystyrene. Dash -dot line: equation (7) for polyethylene terephthalate (PETP). Solid line 1: the present result for the surface dose of PETP. Solid line 2: the present result for the 100-/zm layer of PETP. RPC 33/f~-B
The experiment referred to in this work was carried out for dose-intercomparison. Tanaka et al. (1989) have reported its details. We use here the selected results obtained with six accelerators. The selection was made to adopt varieties of irradiation conditions with reliable values of the effective charge-density q. The absorbed dose was measured with the radiochromic dosimeter (RCD). Sheets of R C D film (1 x 1 cm area) were piled into a stack of thickness larger than the range of incident electrons. The stack was irradiated on a conveyer, and the absorbed dose in the uppermost film was determined. The situation just corresponds to the three-component problem.
5. RESULTS AND DISCUSSION 5.1. Comparison previous results
with approximate
relations and
First, we compared the values of K obtained by E D M U L T with those given by approximate relations. The irradiation conditions assumed are given in Table 1 (the first two rows). The energy region considered is from 100keV to 10MeV. Figure 1 shows the results. The accuracy of the present values of K including those described later is determined by the accuracy of the E D M U L T algorithm. The possible error of the algorithm was estimated to be 8 - 1 0 % from the comparison with experimental and Monte-Carlo results (Tabata and Ito, 1981a). In this comparison, the doses up to the deepest penetration were considered. The multilayers used consisted of materials of widely different atomic numbers. In the evaluation of K, on the other hand, we use only part of the depth-dose curve around the maximum. The atomic numbers of the materials in multilayers assumed are less widely different. In such cases, the error of the algorithm would be less than the stated values. Further, minor errors in the original code were removed in the course of the present work as described in Section 3. I. F r o m these facts, we consider the possible error in the present values of K to be less than 8%.
414
TATSUO TABATA el al. Table 2. Comparison of estimated absorbed-dose De and measured absorbed-dose Dm. The conditions for the window foil and the air layer for the accelerators from A to F were used to draw curves from 1 to 6 i n Fig. 2 Accelerator
Window foil
Air layer (cm)
Energy (keV)
De (kGy)
(kGy)
DiD m
A B C D E F
AI, 30 p m A1, 50 # m Ti, 12.5,um Ti, 12.5 ,urn Ti, 30,um Ti, 45 ,urn
3.5 7.5 5.6 15 8 12
200 300 160 270 220 300
57.2 56.5 49.9 47.8 56.1 49.2
52.9 52.5 45.4 46.3 50.8 48.5
1.08 1.08 1.10 1.03 1.10 1.01
The approximate relations compared in Fig. 1 are the simplest ones often used for rough estimation of the absorbed dose. These can be derived by assuming extreme conditions. When irradiation is made in a vacuum chamber directly connected to the accelerator, the effects of the window foil and the air layer are absent. For high initial energies, the same effects are negligible. In both these cases, we obtain the following two approximations. (i) For x, equal to the extrapolated range of Rex of incident electrons in the sample material. Aside from a minor effect of backscattering, almost all the initial energy is absorbed in the sample. Then we have K ~ 102Eo/Rex (E o in MeV,
Rex
in g/cm2).
(6)
(ii) For x s much smaller than the range of incident electrons in the simple material (the atomic numbers of the sample and the substratum materials are assumed to be low). The function l(x, Eo) in equation (3) at any depth in the sample is approximatley equal to the collision stopping-power S(Eo) (in MeV cm2/g) of the sample material for the electrons of energy E 0. So we have K ~ IO2/S(Eo).
(7)
In Fig. 1, the curve of equation (6) was drawn by using the semiempirical formula of Tabata et al. (1972) for Rex. For the curve of equation (7), use is made of the stopping-power values in ICRU Report 37 (1984). Figure 1 shows the following. (i) For energies below about 500 keV, equation (6) behaves quite differently from the values of K in actual irradiationconditions. (ii) When the window foil and the air layer lie on the sample, equation (7) is a good approximation only for energies above about 2 MeV. (iii) The curves of K versus initial energy obtained in this work show a maximum at energies of 260 and 300 keV. This is due to strong absorption of electrons in the window and the air layer at the lower energies. Next, we compare the values of K obtained by E D M U L T with those reported by the previous authors (Okabe et al., 1974; Tabata, 1975; Proksch et al., 1979). The conditions assumed for curve 2 in Fig. 1 are the same as those assumed by Tabata (1975) (see Table 1). Both Tabata's calculation and the E D M U L T algorithm used the concept of an equivalent layer to treat the passage of electrons through the interface. However, the relations used for
Dm
defining equivalence are different. In spite of this difference, Tabata's results are in complete agreement with the present results except a single value at 300 keV. Okabe et al. (1974) used the depth-dose curve for aluminium (the substratum material) also for the sample layer, and applied no correction. Therefore, we have corrected their value for the ratio of the stopping power of the sample material to that of aluminium. The corrected value is 491 kGym2/C. The present calclation for the same conditions was made by using the method described for the fourcomponent problem. The result is 529 kGy m2/C. The present calculation for the conditions assumed by Proksch et al. (1979) gives 387 kGy m2/C. This is to be compared with the value of Proksch et al., 358 kGy m2/C. In both these cases, the previous result is smaller by 7% than the present value. This deviation is within the possible error of the present calculation. A larger part of the deviation, however, might be attributed to simpler approximations used by the previous authors. 5.2. The three-component problem Values of K were calculated for the sets of conditions of the window and the air layer used in the experiment. The conditions are given in Table 2. The energies considered are from 150 to 350keV. The sample material is nylon, the base material of the RCD. The sample thickness xs assumed is the thickness of a sheet of RCD film, and is 48/~m (5.4 × 10 -3 g/cm2). The results of the calculation are shown in Fig. 2 by solid lines. The dashed line represents the approximation of Tanaka et al. (1979) given by K
~ 102/s(xp,
E0).
(8)
Here Is (x, Eo) is the energy deposition (in MeV cmZ/g electron) by electrons of initial energy E0 at the depth x in the semiinfinite layer of the sample material, and xp is the depth at which I~(x, Eo) reaches the maximum. Figure 2 indicates the following. (i) The curve of equation (8) corresponds roughly to the envelope for the curves obtained by EDMULT. (ii) The present values of K are close to the values of equation (8) for some sets of conditions including the initial energy (see arrows for accelerators A, B and D in Fig. 2a and b). (iii) For the other accelerators, however, there
Absorbed dose and electron-beam processing 800
~
,
,
,
[
,
~
,
r
I
'
'
'
'
[
'
'
T
Table 3. Comparison of the absorbed-dose coefficients K for threecomponent problems calculated by assuming a single composite layer for the window and air layers and those calculated by the regular treatment. Sets of conditions from A to F correspond to those of accelerators from A to F in Table 2
,
600 k-
,T.
K (kGy m-~/C)
u-
i
415
2°° I i-
I
OIi 150
i
l
l
i ~ 200
~
- -
TANAKA
--
PRESENT
i
l
INITIAL
800
"-~.
'
~
J
r
i
r
~
ET
I , 250
i
ENERGY
,
I
i
,
AL.
L
i i 300
r
i
i
E
~
[
r
Composite layer treatment
Regular treatment
Ratio
A B C D E F
672 533 649 574 491 438
669 529 647 574 467 429
1.004 1.008 1.003 1.000 1.05 1.02
r 350
(keY)
"-...
Sets of conditions
i
i
error in De. The error in D m is considered to be less than a few percent. The ratio De/D ~ is greater than unity for all the sets of conditions. The reason for this is not clear.
J
(b)
E
5.3. The four-component problem
600
C U
Ew
4
400
F
8
200
0 150
°,7;7 ...... 200
250 INITIAL
ENERGY
300
350
(keY)
Fig. 2. Absorbed-dose coefficient as a function of initial energy. The sample material assumed is nylon. Present results are for the sample thickness of 48 #m. The window foil and the air layer assumed for each curve are given in Table 2. (a) Aluminium window, (b) titanium window.
are deviations ranging from 19 to 34%. In short, equation (8) gives a good estimate in some cases, and an overestimation in others. The nominal energy of accelerator E used in the experiment was 300 keV. At this energy, the agreement of equation (8) with the value of K given by E D M U L T is satisfying (see Fig. 2b). From the depth-dose curve obtained by the RCD, however, the actual energy was estimated to be about 220 keV. For this energy, the window and air layers used are too thick, and absorb about 50% of the initial energy. Therefore, the set of conditions used by accelerator E is not the typical of practical irradition. From the values of K and the measured values of q, absorbed doses in the RCD irradiated with the six accelerators were estimated. The estimated values D~ are compared with the measured values D,, in Table 2. The maximum deviation of D~ from Dm is 10% (see the last column of Table 2). The r.m.s, deviation is 6.7%. Considering from possible errors in D~ and Din, this agreement is reasonably good. In addition to the error in K, appreciable uncertainties in the beam current i0 and the beam width w contribute to the
Table 3 shows the result of check on the validity of the composite-layer treatment described in Section 3.3. Except for set E, this treatment gives the value of K agreeing with the result of the regular treatment within 2%. Therefore, we conclude that the composite-layer treatment can be used for practical conditions without appreciably increasing the error in K. By using the composite-layer treatment, depthdose distributions in the sample layer were calculated for different substratum materials. The conditions assumed were the same as those used in one of the calculations by Entinzon (1986) except the initial energy of electrons. Entinzon assumed a broad spectrum extending from 0 to 350 keV. In the present calculation, a monoenergetic beam of 300-keV energy was assumed. The window was a titanium foil of 30-/~m thickness. The air layer was 10 cm thick. The sample consisted of cellulose triacetate (CTA), and was 170/~m (22-mg/cm2) thick. Substratum materials were CTA, aluminium, germanium, molybdenum and tungsten. The results are shown in Fig. 3. The dose in the CTA layer increases with increasing atomic number of the subtratum material similarly to Entinzon's result. This is due to the backscattering of electrons from the substratum. The discontinuities of the dose at the interfaces are caused by the discontinuous change in the stopping power of the medium. Cusps on curves 3-5 and on the curve in the substratum region have been caused by the single-layer algorithm repeatedly used in EDMULT. This algorithm joins two functional forms at a large depth, and a cusp appears at this joint. The areas under curves 1-5 divided by the thickness of the CTA layer give values of K. The values corresponding to curves 1-5 are 440, 465, 519, 539 and 584 kGy m2/C. The backscattering from the tungsten substratum increases the value of K by 33% compared with the case of the CTA substratum.
416
TATSUO TABATA et al. G E
I
600
'
I
I
lAIR] CTA
'
I
'
SUBSTRATUM
~ 400
200
0
i
40
80
120
THICKNESS (mg/cm 2)
Fig. 3. Depth~lose distributions in the CTA layer placed on various substratum materials. The incident energy is 300 keV. Curves in the regions of the titanium, air and substratum layers show the distributions when the substratum is CTA. The substratum for curve (1) CTA; (2) aluminium; (3) germanium; (4) molybdenum; (5) tungsten.
Entinzon I. R. (1986) Radiat. Phys. Chem. 27, 469. Halbleib J. A. and Mehlhorn T. A. (1984) Sandia Natl Labs Rep. SAND 84-0573. International Commission on Radiation Units and Measurements (1984) Stopping Powers for Electrons and Positrons, ICRU Rep. 37, ICRU, Bethesda, MD. Lax I. and Brahme A. (1985) Acta radiol. Oncol. 24, 75. Markus B. (1961) Strahlentherapie 116, 280. Okabe S., Tsumori K., Tabata T., Yoshida T., Nagai A., Hiro S., Ishida K., Sakamoto I., Kawai T. Arakawa K., Inoue T. and Murakami T. (1974) Oyo Buturi 43, 909. Proksch E., Gehringer P. and Eschweiler H. (1979) Int. J. appl. Radiat. Isot. 30, 279. Rao B. N. S. (1966) Nucl. Instrum. Meth. 44, 155. Tabata T. (1975) Ionizing Radiat. 1(4), 49. Tabata T. and Ito R. (1981a) Jpn J. appl. Phys. 20, 249. Tabata T. and Ito R. (1981b) Radiat. Center Osaka Prefect. Tech. Rep. No. 1; ibid. (1983) No. 3, Tabata T. and Ito R. (1982) RSIC Computer Code Collection CCC-430, Radiat. Shielding Information Center, Oak Ridge Natl Lab. Tabata T., lto R. and Okabe S. (1972) Nucl. lnstrum. Meth. 103, 85. Tanaka R., Sunaga H., Kuriyama I. and Moriuchi Y. (1989) Radiat. Phys. Chem.
6. C O N C L U S I O N
T h e present work has s h o w n the following. (i) T h e a b s o r b e d dose coefficients K for the three- a n d fourc o m p o n e n t p r o b l e m s can rapidly be evaluated by using the code E D M U L T o n a personal computer. (ii) The values o f K o b t a i n e d by E D M U L T agree reasonably well with those reported by the previous authors. (iii) The a d s o r b e d doses estimated from K show m o d e r a t e a g r e e m e n t with the values m e a s u r e d by the r a d i o c h r o m i c dosimeter for various sets of irradiation conditions. (iv) A p p r o x i m a t e relations for K are applicable only to limited conditions.
APPENDIX In this Appendix, another expression for the effective atomic-number of a compound or a mixture is given. The reason why we used equation (4) is also given. Equation (4) was originally proposed as the expression for the effective atomic-number to be used in the range-energy relation (Markus, 1961; Rao, 1966). Recently Lux and Brahme (1985) have reported that the following relation is better for depth~lose curves:
/
= E n,Z~/2 n,Z,, i
REFERENCES
Berger M. J. and Seltzer S. M. (1969a) Quality of radiation in a Water Medium Irradiated with High-Energy Electron Beams. Presented at the 12th Int. Congr. of Radiology, Tokyo, Natl. Bureau of Standards (Unpublished). Berger M. J. and Seltzer S. M. (1969b) Ann. Acad. Sci. N.Y. 161, 8, Berger M. J. and Seltzer S. M. (1969c) RSIC Computer Code Collection CCC-107, Radiat. Shielding Information Center Oak Ridge Natl Lab.
/
(Al)
i
where n~ is the number fraction of the constituent element i. Using equations (4) and (A1) in EDMULT, we have calculated depths:lose distributions of electrons in water. Results (Berger and Seltzer, 1969a,b) obtained by the Monte Carlo code ETRAN (Berger and Seltzer, 1969c) served as a standard. Comparison showed the following. At energies of 10 and 20 MeV, equation (A1) gives better result around the maximum of the distribution. At a lower energy of 1 MeV, however, the result obtained with equation (AI) is worse. Therefore, we used equation (4) in the present work.
Radiat. Phys. Chem. Vol. 33, No. 5, pp. 411~t16, 1989 Int. J. Radiat. Appl. Instrum. Part C
Printed in Great Britain. All rights reserved
SIMPLE METHOD OF EVALUATING ABSORBED DOSE IN ELECTRON-BEAM PROCESSINGt TATSUO TABATA,1 RINSUKEITO,1 ISAMU KURIYAMA2 and YASUYUKIMORIUCHI3 'Radiation Center of Osaka Prefecture, Sakai, Osaka 593, 2Takasaki Branch, Irradiation Development Association, Takasaki, Gunma 370-12 and 3Gifu College of Medical Technology, Seki, Gifu 501-32, Japan Abstract--The use of semiempirical multilayer depth-dose code EDMULT for evaluating the absorbeddose coefficient K was studied with a personal computer. When multiplied by the charge per unit area, the coefficient gives the adsorbed dose in the sample irradiated by electrons. This relation presupposes the model configuration of the plane-parallel beam normally incident on the four-layer slab absorber. The absorber consists of the accelerator window, the air layer, the sample and the substratum. The initial energies of electrons above 100 keV were considered. The values of K obtained agreed reasonbly well with the values in the literature. The absorbed doses estimated from K were compared with the values measured by the radiochromic dosimeter for various sets of irradiation conditions. The r.m.s, deviation of the former from the latter was 6.7%. A calculated example of the effects of different substratum materials is given.
l. INTRODUCTION In planning the radiation processing with the lowenergy electron beam, it is important to estimate the absorbed dose in the sample. Electrons suffer appreciable energy-loss and scattering in the accelerator window and an air layer before entering the sample. Under the sample, there is often a substratum consisting of a metal, which has a considerable backscattering effect. Therefore, we have to treat the transport problem of electrons through a multilayer absorber with three or four components. MonteCarlo code systems to solve such problems are available for general use; examples are Integrated T I G E R Series (ITS) codes (Halbleib and Mellhorn, 1984). Recently Entinzon (Entinzon, 1986) has developed a method to solve the energy-transport equation numerically, and has evaluated dose distributions in multilayer absorbers. Both the Monte-Carlo codes and the numerical method, however, require a considerable time on a large computer. Therefore, it is desirable to develop a simple and accurate method for estimating the absorbed dose. Okabe et al. (Okabe et al., 1974) have proposed a simple relation to evaluate the absorbed dose for the standard geometry of electron irradiation. Their relation gives the absorbed dose as the product of the charge per unit area and a coefficient K. They have estimated the value of K for 300-keV electrons. Tabata (1975) has also given values of K for the fThis work was performed as an activity of Low-Energy Electron-Dose Measurement Committee sponsored by Irradiation Development Association. A preliminary account of part of this work was given at the Conference on Radiation Curing Asia, Tokyo, October 1986.
energies between 300 keV and 2 MeV. Proksch et al. (1979) have given a value of a coefficient, which can be reduced to K, at 500 keV. Tanaka et al. (1989) have proposed an approximate relation for K applicable to conditions of practical irradiation with lowenergy electrons. In this work, we studied a method to rapidly evaluate K for the sample in a multilayer of three or four components. A semiempirical three-layer depth-dose code EDMULT developed by Tabata and Ito (1981a, b, 1982, 1983) was used on a personal computer. Accuracy was checked by comparing (i) values of K with those reported by previous authors (Okabe et al., 1974; Tabata, 1975; Proksch et al., 1979) and (ii) the absorbed doses estimated from K, with measured values for various sets of irradiation conditions. 2. FORMULATION OF THE PROBLEM
We consider a model configuration for the standard irradiation on a conveyer with the low-energy electron accelrator. The configuration, originally used by Okabe et al. (1974) is as follows: the planeparallel beam is normally incident on a four-layer slab absorber. The first layer is the accelerator window made of aluminium or titanium foil of thickness x,.. The second is the air layer of thickness x a. The third is the sample layer of thickness x,. The fourth is the substratum of enough thickness to stop all the incident electrons (effectively semiinfinite thickness). This configuration is equivalent to the actual one for the accelerators of linear-filament type and broadbeam type. It is also a good approximation for the accelerator of scanning-beam type when the maximum angle of scanning is small.
411
412
TATSUOTABATAet al.
Our problem is to evaluate the absorbed dose D averaged over the thickness x~ of the sample layer. We treat the two cases of substratum materials (i) the same as and (ii) different from the sample material. The first case occurs, for example, when the absorbed dose in the uppermost sheets in a thick stack of film is considered. We call case (i) the three-component problem, and case (ii) the four-component problem. The following quantities are usually known: i0 = beam current in the accelerator tube (in A); v = conveyer speed (in m/s); w = beam width perpendicular to the movement of the conveyer at the sample surface (in m); E 0 -- initial energy of electrons when incident on the window (known from the accelerating voltage). From the first three quantities, we define effective charge-density q (charge per unit area) by: q = io/vw (in C/m2).
(1)
The product vw corresponds to the area of the irradiation field divided by the irradiation time. While we treat a multilayer including the window, we used here this product evaluated at the sample surface. The reason is as follows. In the depth-dose code EDMULT we used, the irradiation field assumed is of infinite area. So this code does not include the broadening of the field with increasing depth caused by scattering of electrons in the medium. To use the code for a case in which the broadening effect is appreciable, therefore, we must take it into account somehow. We can do this by using the field area at the depth in the absorber where dose is to be evaluated. This is equal to the field area at the sample surface (the linear thickness of the sample is usually negligible compared with that of the air layer). Using q, we can express the absorbed dose D by (Okabe et al., 1974): D = Kq
(in kGy)
(2)
Here K is defined by I i ~ + x a + x~
K = (102/x~)
l(x, Eo) dx ~¢+ Xa
(in kGy m2/C).
(3)
We call it absorbed-dose coefficient. In equation (3), I(x, Eo) is the energy deposition (in MeV cm2/g electron) by the electrons of initial energy E0 at the depth x in the multilayer absorber. The factor 102 stands for the arrangement of the units. Aside from this factor, K is equal to the absorbed energy per unit thickness per incident electron, averaged over the sample thickness. In the present model, the problem of evaluating the absorbed dose is thus divided into the problems of evaluating q and K. The present work treats the evaluation of K. As for the evaluation of q, the following must be noted. In some accelerators, the
pipe for coolant attached to the outside of the window intercepts a fraction of the beam. Then we must use as i0 the beam current in the accelerator tube reduced by the intercepted fraction. It is also necessary to take into account the gradient of the current density at the edges of the irradiation field in evaluating w. This can be done by using the "half-width" of the current-density distribution. 3. C O M P U T A T I O N A L
METHOD
3.1. The code used
We used the code EDMULT to compute the absorbed-dose coefficient K. This code was developed to evaluate depth~lose distributions produced by plane-parallel electron beams normally incident on multilayer slab absorbers (Tabata and Ito, 1981a,b, 1982, 1983). The numbers of layers permitted are from one to three. The last layer in the three-layer absorber must be effectively semiinfinite. The information obtainable are differential and integral energy-depositions as a function of depth in the absorber. The code is applicable to the initial energies from 0.1 to 20 MeV and to almost all materials including mixtures and compounds. The algorithm of the code uses semiemipirical realtions, and most of the calculations are made analytically. In the course of the present work, minor errors were found in the original code, and were removed. This improved agreement with experimental and Monte-Carlo results (cited in Tabata and Ito, 1981a) at the smallest depths in the third layer. Doses at these depths are most important in evaluating K for the three-component problems. The code EDMULT, originally written for the large computer, was adapted to a personal computer. An application program of EDMULT was developed to compute K and depth-dose distributions. The majority of subprograms are coded in F O R T R A N 77. The program can run on the operating system MS-DOS version 3.10 without modification for NEC PC-9800 series machines. Use of the machine with or without a numerical coprocessor is permitted. When the machine is equipped with a numerical coprocessor, the computation time required is quite small (a few seconds for a value of K). A floppy-disk copy of this program is available on request. 3.2. The three-component problem
The configuration of the three-component problem is the same as that assumed in EDMULT. In this problem, therefore, application of E D M U L T to the computation of K is simple. To evaluate the integral in equation (3), we call EDMULT twice for the depths corresponding to the limits of the integral. For the atomic number Z and the atomic weight A of air, the effective values given by the following formulas are used: Z = ~ f Z,; i
(4)
Absorbed dose and electron-beam processing
413
Table 1. Conditions assumed in evaluating the absorbed-dose coefficient K shown in Fig. I. The symbols x,., x~ and x, represent the thickness of the window foil, the air layer and the sample layer, respectively. For x~= 0, the value of K gives the surface dose when used in equation (2). The substratum material assumed is the same as the sample material except Okabe et al. (1974) who assumed aluminium x.. xa Sample x~ Author (,urn) (cm) material (#m) Present work (1) Present work (2) Okabe et al. (1974) Tabata (1975) Proksch et al. (1979)b
30 30 30 30 25
8 8 8 8 22
PETI~ PETP PETP PETP Nylon
0 100 100 100 0
~Polyethylene terephthalate, bAngle of incidence was assumed to be 15'>.
4. EXPERIMENTAL METHOD
Here f~ is the fraction by weight of the constituent element with atomic number Z~ and atomic weight A i. We use these equations also for the sample material consisting of a c o m p o u n d or a mixture. A different definition of the effective atomic number is described in the Appendix. 3.3. The f o u r - c o m p o n e n t problem
To treat the four-component problem by EDM U L T , we consider to reduce it to the threecomponent problem. We do this by treating the window and the air layer as a single composite-layer. The effective atomic number and atomic weight for the composite layer can be calculated again from equations (4) and (5). In this use of these equations, the fraction by weight f is equal to the mass-thickness fraction. To check the validity of this treatment, we compared the values of K for the three-component problems with those obtained by the regular treatment. The result of the comparison is described in Section 5.3. k
6ec
'
I , ,,r
>.
l',/""\ '/x ~ I \ \L
,oo
I
F
×
OKABE
•
TAB,
o .....
o
'
I
' ''
ET AL.
,,
PROKSCN ET , L
EO.
Ce~
"' PRESENT
20C
o m 0.1
0.5 INITIAL
I ENERGY
5
I0
(MeV1
Fig. 1. Absorbed-dose coefficient as a function of initial energy. The conditions assumed in the previous and present evaluations are given in Table 1. Dashed line: equation (6) for polystyrene. Dash -dot line: equation (7) for polyethylene terephthalate (PETP). Solid line 1: the present result for the surface dose of PETP. Solid line 2: the present result for the 100-/zm layer of PETP. RPC 33/f~-B
The experiment referred to in this work was carried out for dose-intercomparison. Tanaka et al. (1989) have reported its details. We use here the selected results obtained with six accelerators. The selection was made to adopt varieties of irradiation conditions with reliable values of the effective charge-density q. The absorbed dose was measured with the radiochromic dosimeter (RCD). Sheets of R C D film (1 x 1 cm area) were piled into a stack of thickness larger than the range of incident electrons. The stack was irradiated on a conveyer, and the absorbed dose in the uppermost film was determined. The situation just corresponds to the three-component problem.
5. RESULTS AND DISCUSSION 5.1. Comparison previous results
with approximate
relations and
First, we compared the values of K obtained by E D M U L T with those given by approximate relations. The irradiation conditions assumed are given in Table 1 (the first two rows). The energy region considered is from 100keV to 10MeV. Figure 1 shows the results. The accuracy of the present values of K including those described later is determined by the accuracy of the E D M U L T algorithm. The possible error of the algorithm was estimated to be 8 - 1 0 % from the comparison with experimental and Monte-Carlo results (Tabata and Ito, 1981a). In this comparison, the doses up to the deepest penetration were considered. The multilayers used consisted of materials of widely different atomic numbers. In the evaluation of K, on the other hand, we use only part of the depth-dose curve around the maximum. The atomic numbers of the materials in multilayers assumed are less widely different. In such cases, the error of the algorithm would be less than the stated values. Further, minor errors in the original code were removed in the course of the present work as described in Section 3. I. F r o m these facts, we consider the possible error in the present values of K to be less than 8%.
414
TATSUO TABATA el al. Table 2. Comparison of estimated absorbed-dose De and measured absorbed-dose Dm. The conditions for the window foil and the air layer for the accelerators from A to F were used to draw curves from 1 to 6 i n Fig. 2 Accelerator
Window foil
Air layer (cm)
Energy (keV)
De (kGy)
(kGy)
DiD m
A B C D E F
AI, 30 p m A1, 50 # m Ti, 12.5,um Ti, 12.5 ,urn Ti, 30,um Ti, 45 ,urn
3.5 7.5 5.6 15 8 12
200 300 160 270 220 300
57.2 56.5 49.9 47.8 56.1 49.2
52.9 52.5 45.4 46.3 50.8 48.5
1.08 1.08 1.10 1.03 1.10 1.01
The approximate relations compared in Fig. 1 are the simplest ones often used for rough estimation of the absorbed dose. These can be derived by assuming extreme conditions. When irradiation is made in a vacuum chamber directly connected to the accelerator, the effects of the window foil and the air layer are absent. For high initial energies, the same effects are negligible. In both these cases, we obtain the following two approximations. (i) For x, equal to the extrapolated range of Rex of incident electrons in the sample material. Aside from a minor effect of backscattering, almost all the initial energy is absorbed in the sample. Then we have K ~ 102Eo/Rex (E o in MeV,
Rex
in g/cm2).
(6)
(ii) For x s much smaller than the range of incident electrons in the simple material (the atomic numbers of the sample and the substratum materials are assumed to be low). The function l(x, Eo) in equation (3) at any depth in the sample is approximatley equal to the collision stopping-power S(Eo) (in MeV cm2/g) of the sample material for the electrons of energy E 0. So we have K ~ IO2/S(Eo).
(7)
In Fig. 1, the curve of equation (6) was drawn by using the semiempirical formula of Tabata et al. (1972) for Rex. For the curve of equation (7), use is made of the stopping-power values in ICRU Report 37 (1984). Figure 1 shows the following. (i) For energies below about 500 keV, equation (6) behaves quite differently from the values of K in actual irradiationconditions. (ii) When the window foil and the air layer lie on the sample, equation (7) is a good approximation only for energies above about 2 MeV. (iii) The curves of K versus initial energy obtained in this work show a maximum at energies of 260 and 300 keV. This is due to strong absorption of electrons in the window and the air layer at the lower energies. Next, we compare the values of K obtained by E D M U L T with those reported by the previous authors (Okabe et al., 1974; Tabata, 1975; Proksch et al., 1979). The conditions assumed for curve 2 in Fig. 1 are the same as those assumed by Tabata (1975) (see Table 1). Both Tabata's calculation and the E D M U L T algorithm used the concept of an equivalent layer to treat the passage of electrons through the interface. However, the relations used for
Dm
defining equivalence are different. In spite of this difference, Tabata's results are in complete agreement with the present results except a single value at 300 keV. Okabe et al. (1974) used the depth-dose curve for aluminium (the substratum material) also for the sample layer, and applied no correction. Therefore, we have corrected their value for the ratio of the stopping power of the sample material to that of aluminium. The corrected value is 491 kGym2/C. The present calclation for the same conditions was made by using the method described for the fourcomponent problem. The result is 529 kGy m2/C. The present calculation for the conditions assumed by Proksch et al. (1979) gives 387 kGy m2/C. This is to be compared with the value of Proksch et al., 358 kGy m2/C. In both these cases, the previous result is smaller by 7% than the present value. This deviation is within the possible error of the present calculation. A larger part of the deviation, however, might be attributed to simpler approximations used by the previous authors. 5.2. The three-component problem Values of K were calculated for the sets of conditions of the window and the air layer used in the experiment. The conditions are given in Table 2. The energies considered are from 150 to 350keV. The sample material is nylon, the base material of the RCD. The sample thickness xs assumed is the thickness of a sheet of RCD film, and is 48/~m (5.4 × 10 -3 g/cm2). The results of the calculation are shown in Fig. 2 by solid lines. The dashed line represents the approximation of Tanaka et al. (1979) given by K
~ 102/s(xp,
E0).
(8)
Here Is (x, Eo) is the energy deposition (in MeV cmZ/g electron) by electrons of initial energy E0 at the depth x in the semiinfinite layer of the sample material, and xp is the depth at which I~(x, Eo) reaches the maximum. Figure 2 indicates the following. (i) The curve of equation (8) corresponds roughly to the envelope for the curves obtained by EDMULT. (ii) The present values of K are close to the values of equation (8) for some sets of conditions including the initial energy (see arrows for accelerators A, B and D in Fig. 2a and b). (iii) For the other accelerators, however, there
Absorbed dose and electron-beam processing 800
~
,
,
,
[
,
~
,
r
I
'
'
'
'
[
'
'
T
Table 3. Comparison of the absorbed-dose coefficients K for threecomponent problems calculated by assuming a single composite layer for the window and air layers and those calculated by the regular treatment. Sets of conditions from A to F correspond to those of accelerators from A to F in Table 2
,
600 k-
,T.
K (kGy m-~/C)
u-
i
415
2°° I i-
I
OIi 150
i
l
l
i ~ 200
~
- -
TANAKA
--
PRESENT
i
l
INITIAL
800
"-~.
'
~
J
r
i
r
~
ET
I , 250
i
ENERGY
,
I
i
,
AL.
L
i i 300
r
i
i
E
~
[
r
Composite layer treatment
Regular treatment
Ratio
A B C D E F
672 533 649 574 491 438
669 529 647 574 467 429
1.004 1.008 1.003 1.000 1.05 1.02
r 350
(keY)
"-...
Sets of conditions
i
i
error in De. The error in D m is considered to be less than a few percent. The ratio De/D ~ is greater than unity for all the sets of conditions. The reason for this is not clear.
J
(b)
E
5.3. The four-component problem
600
C U
Ew
4
400
F
8
200
0 150
°,7;7 ...... 200
250 INITIAL
ENERGY
300
350
(keY)
Fig. 2. Absorbed-dose coefficient as a function of initial energy. The sample material assumed is nylon. Present results are for the sample thickness of 48 #m. The window foil and the air layer assumed for each curve are given in Table 2. (a) Aluminium window, (b) titanium window.
are deviations ranging from 19 to 34%. In short, equation (8) gives a good estimate in some cases, and an overestimation in others. The nominal energy of accelerator E used in the experiment was 300 keV. At this energy, the agreement of equation (8) with the value of K given by E D M U L T is satisfying (see Fig. 2b). From the depth-dose curve obtained by the RCD, however, the actual energy was estimated to be about 220 keV. For this energy, the window and air layers used are too thick, and absorb about 50% of the initial energy. Therefore, the set of conditions used by accelerator E is not the typical of practical irradition. From the values of K and the measured values of q, absorbed doses in the RCD irradiated with the six accelerators were estimated. The estimated values D~ are compared with the measured values D,, in Table 2. The maximum deviation of D~ from Dm is 10% (see the last column of Table 2). The r.m.s, deviation is 6.7%. Considering from possible errors in D~ and Din, this agreement is reasonably good. In addition to the error in K, appreciable uncertainties in the beam current i0 and the beam width w contribute to the
Table 3 shows the result of check on the validity of the composite-layer treatment described in Section 3.3. Except for set E, this treatment gives the value of K agreeing with the result of the regular treatment within 2%. Therefore, we conclude that the composite-layer treatment can be used for practical conditions without appreciably increasing the error in K. By using the composite-layer treatment, depthdose distributions in the sample layer were calculated for different substratum materials. The conditions assumed were the same as those used in one of the calculations by Entinzon (1986) except the initial energy of electrons. Entinzon assumed a broad spectrum extending from 0 to 350 keV. In the present calculation, a monoenergetic beam of 300-keV energy was assumed. The window was a titanium foil of 30-/~m thickness. The air layer was 10 cm thick. The sample consisted of cellulose triacetate (CTA), and was 170/~m (22-mg/cm2) thick. Substratum materials were CTA, aluminium, germanium, molybdenum and tungsten. The results are shown in Fig. 3. The dose in the CTA layer increases with increasing atomic number of the subtratum material similarly to Entinzon's result. This is due to the backscattering of electrons from the substratum. The discontinuities of the dose at the interfaces are caused by the discontinuous change in the stopping power of the medium. Cusps on curves 3-5 and on the curve in the substratum region have been caused by the single-layer algorithm repeatedly used in EDMULT. This algorithm joins two functional forms at a large depth, and a cusp appears at this joint. The areas under curves 1-5 divided by the thickness of the CTA layer give values of K. The values corresponding to curves 1-5 are 440, 465, 519, 539 and 584 kGy m2/C. The backscattering from the tungsten substratum increases the value of K by 33% compared with the case of the CTA substratum.
416
TATSUO TABATA et al. G E
I
600
'
I
I
lAIR] CTA
'
I
'
SUBSTRATUM
~ 400
200
0
i
40
80
120
THICKNESS (mg/cm 2)
Fig. 3. Depth~lose distributions in the CTA layer placed on various substratum materials. The incident energy is 300 keV. Curves in the regions of the titanium, air and substratum layers show the distributions when the substratum is CTA. The substratum for curve (1) CTA; (2) aluminium; (3) germanium; (4) molybdenum; (5) tungsten.
Entinzon I. R. (1986) Radiat. Phys. Chem. 27, 469. Halbleib J. A. and Mehlhorn T. A. (1984) Sandia Natl Labs Rep. SAND 84-0573. International Commission on Radiation Units and Measurements (1984) Stopping Powers for Electrons and Positrons, ICRU Rep. 37, ICRU, Bethesda, MD. Lax I. and Brahme A. (1985) Acta radiol. Oncol. 24, 75. Markus B. (1961) Strahlentherapie 116, 280. Okabe S., Tsumori K., Tabata T., Yoshida T., Nagai A., Hiro S., Ishida K., Sakamoto I., Kawai T. Arakawa K., Inoue T. and Murakami T. (1974) Oyo Buturi 43, 909. Proksch E., Gehringer P. and Eschweiler H. (1979) Int. J. appl. Radiat. Isot. 30, 279. Rao B. N. S. (1966) Nucl. Instrum. Meth. 44, 155. Tabata T. (1975) Ionizing Radiat. 1(4), 49. Tabata T. and Ito R. (1981a) Jpn J. appl. Phys. 20, 249. Tabata T. and Ito R. (1981b) Radiat. Center Osaka Prefect. Tech. Rep. No. 1; ibid. (1983) No. 3, Tabata T. and Ito R. (1982) RSIC Computer Code Collection CCC-430, Radiat. Shielding Information Center, Oak Ridge Natl Lab. Tabata T., lto R. and Okabe S. (1972) Nucl. lnstrum. Meth. 103, 85. Tanaka R., Sunaga H., Kuriyama I. and Moriuchi Y. (1989) Radiat. Phys. Chem.
6. C O N C L U S I O N
T h e present work has s h o w n the following. (i) T h e a b s o r b e d dose coefficients K for the three- a n d fourc o m p o n e n t p r o b l e m s can rapidly be evaluated by using the code E D M U L T o n a personal computer. (ii) The values o f K o b t a i n e d by E D M U L T agree reasonably well with those reported by the previous authors. (iii) The a d s o r b e d doses estimated from K show m o d e r a t e a g r e e m e n t with the values m e a s u r e d by the r a d i o c h r o m i c dosimeter for various sets of irradiation conditions. (iv) A p p r o x i m a t e relations for K are applicable only to limited conditions.
APPENDIX In this Appendix, another expression for the effective atomic-number of a compound or a mixture is given. The reason why we used equation (4) is also given. Equation (4) was originally proposed as the expression for the effective atomic-number to be used in the range-energy relation (Markus, 1961; Rao, 1966). Recently Lux and Brahme (1985) have reported that the following relation is better for depth~lose curves:
/
= E n,Z~/2 n,Z,, i
REFERENCES
Berger M. J. and Seltzer S. M. (1969a) Quality of radiation in a Water Medium Irradiated with High-Energy Electron Beams. Presented at the 12th Int. Congr. of Radiology, Tokyo, Natl. Bureau of Standards (Unpublished). Berger M. J. and Seltzer S. M. (1969b) Ann. Acad. Sci. N.Y. 161, 8, Berger M. J. and Seltzer S. M. (1969c) RSIC Computer Code Collection CCC-107, Radiat. Shielding Information Center Oak Ridge Natl Lab.
/
(Al)
i
where n~ is the number fraction of the constituent element i. Using equations (4) and (A1) in EDMULT, we have calculated depths:lose distributions of electrons in water. Results (Berger and Seltzer, 1969a,b) obtained by the Monte Carlo code ETRAN (Berger and Seltzer, 1969c) served as a standard. Comparison showed the following. At energies of 10 and 20 MeV, equation (A1) gives better result around the maximum of the distribution. At a lower energy of 1 MeV, however, the result obtained with equation (AI) is worse. Therefore, we used equation (4) in the present work.