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Calculation of absorbed dose of low-energy electron beam by an approximate method
Radiation Physics and Chemistry 64 (2002) 181–187
Calculation of absorbed dose of low-energy electron beam by an approximate method ! G. Va! zquez-Poloa,*, H. Lopez Valdiviab, H. Carrasco Abregob, R.R. Mijangosc, R. Garc!ıa Ga ! Instituto de F!ısica, UNAM, AP 20-364, Mexico, DF 01000, Mexico ! Instituto Nacional de Investigaciones Nucleares, AP 18-1027, Mexico, DF 11801, Mexico c ! en F!ısica UNISON, AP 5-88.83190, Hermosillo, Son. Mexico Centro de Investigacion a
b
Received 6 April 2000; accepted 4 July 2001
Abstract In this work, we present a simple method to calculate absorbed doses of low-energy electron beams together with an experimental procedure for calibrating the accelerator under certain work conditions. The semi-empirical method is useful for a rapid estimation of depth–dose curve of electron beams, which uses an adjustable parameter factor determined by the least-squares fit from experimental values. Our approach is tested in electron processes on different materials, obtaining reasonable agreement. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Radiation processing; Electron beam; Depth–dose curve; Algorithm
1. Introduction Electron accelerators have a wide range of applications in the industry and in the basic research. Two of the most common problems during material irradiation with electron beams are determining the absorbed dose and the reached depth. The approach in each case depends on the kind of irradiator, the technological resources and the infrastructure available. See McLaughlin et al. (1975) for a review of this topic. Most of the theoretical work for evaluating the absorbed dose in an irradiated material uses computational methods. Monte Carlo simulation and direct numerical solution of transport equation are some useful methods. However, both of them require large CPU time and make necessary the use of high capacity computers. In this work, we present a semi-empirical method using a standard personal computer for *Corresponding author. Fax: +52-5-622-5011. E-mail address: [email protected]fisicacu.unam.mx (G. V!azquez-Polo).
determining points of the depth–dose curve of monoenergetic electrons with an energy range of 0.1–1.0 MeV. The obtained values agree within an acceptable range, with the experimental measurements. This agreement allows us to have a general experimental process to calibrate low-energy electron accelerators. We are making use of this process in the pelletron-type accelerator at the Instituto Nacional de Investigaciones Nucleares (ININ). In irradiation processing, by employing a beam of low-energy electrons, the geometry of the system can be considerably simplified, following the suggestion of Okabe et al. (1974) and Tabata et al. (1989), where essentially a normally incident beam of electrons collides with a three-layer slab absorber: 1. the accelerator window (titanium foil), 2. the air layer and 3. the sample layer. This geometry is an applicable arrangement for a large majority of electron accelerators of the scanningbeam type where the maximum angle of scanning is small.
0969-806X/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 9 - 8 0 6 X ( 0 1 ) 0 0 4 5 9 - 5
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182
2. Electron beam characterization To adequately calibrate an accelerator, it is necessary to know the profile and dimensions of the electron beam, besides the current intensity. These parameters can be evaluated by the following procedure. Two plates of PVC are exposed to the electron beam in order to have a permanent and visible mark in the sample. These plates are placed, one on site A; at a distance XA from the titanium foil window. The other one is placed in point B covering the aluminum curtain at a distance XB ; as shown in Fig. 1. The marks over the plates produced by the electron beam can be digitized in order to have a better knowledge of the real shape and dimensions of the electron beam in this region. In Fig. 2, we can see the digitized spots of the PVC plates and the spots showing their real size are also visible in Figs. 3a and b, respectively, from which an effective spot radius rA and rB can be evaluated. Using these measurements, it is possible to calculate the effective density of the current and the maximum irradiation area in the sample. For these goals, a virtual point focus located in F at a distance d from the titanium window, is geometrically defined as shown in Fig. 4 (the thickness of the titanium windows is most likely about 40 mm) This distance due to the geometry involved is given by d¼
X B rA X A rB : rB rA
ð1Þ
If the sample is located at distance Xm from the titanium window, the radius of the area is given by rM ¼
rB ðd þ Xm Þ d þ XB
ð2Þ
and the area Am covered by the electron beam over the sample can be calculated. The density of the current of the electron beam that strikes the sample, supposing that
Fig. 2. Digitized spots of the marks over the PVC plates produced by the electron beam.
the beam is homogeneous and monoenergetic is given by I¼
I0 ; Am
ð3Þ
where I0 is the current, in microamperes, measured on the aluminum screen and Am the irradiated area in the sample, given in cm2.
3. Adjustment factor for energy-depth determination In order to evaluate the energy loss of the electron beam when it passes through a given material of thickness DX; the following general expression can be used: DE ¼ E0 Ef ¼ ðdE=dXÞE0 r DX;
ð4Þ
where DE is the kinetic energy lost by the electron beam (MeV), E0 the kinetic energy of the incident electron, before hitting the titanium window (Mev) in this case the profile is 2.4 2.8 cm2 of area, Ef the kinetic energy of the transmitted electron (MeV), ðdE=dXÞE0 the stopping power of the sample for electrons with initial energy E0 (MeV cm2/g), r the density of the material (g/cm3), and DX the thickness in cm penetrated by the electron. From Eq. (4), we get
Fig. 1. View of the electron beam accelerator showing the position of: focus (F), scanner’s windows (V), PVC plates (A, B) and sample (C).
DE DX ¼ : ðdE=dXÞE0 r
ð5Þ
From the experimental data of Nelms (1956), it can be seen that the incremental depth penetration in the
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183
Fig. 4. Virtual focus point (F) created geometrically.
Fig. 3. Digitized spots of the PVC plates in A and B positions (Fig. 1), respectively.
considered energy range is proportional to the electron’s energy loss DE: Now we divide the penetrated sample into DXn segments using for each one ðdE=dXÞn shown in the tables given by Berger and Seltzer (1982) for the corresponding energy DEn of the segments, where DEn is the fraction of the incident energy E0 ; then we can sum these thickness segments up to energy E0 and we then get X¼
nX ¼N n¼1
DXn ¼
X
DEn ðdE=dxÞn
r:
ð6Þ
Eq. (6) can be evaluated for different energies, in the range 0.1 to 1.0 MeV plot X ¼ XE0 as a function of E0 using the electron stopping power values of the National
Bureau of Standards reported by Berger and Seltzer (1982) which take into account: (a) the ionization and excitation losses (collision stopping power) and (b) the bremsstrahlung losses (radiation stopping powers). In a strict sense, this is, still, only an approximation to the theoretical predictions, because we have assumed that the energy of the electrons inside each thickness is homogenous and the exit energy is Ef ¼ E0 DE; as it was assumed in Eq. (4). Additionally, for each value of electron energy inside the material, we must consider the corresponding stopping power value. Taking into account this fact, we modify expression (5) by replacing the interaction between the electron beam and the target with a parameter, which considers the average values of the stopping power factor inside the materials as follows. We propose to adjust the experimental values taken from Nelms (1956), by means of least-square fitting, and relating their slope with the approximated values which gives an adjustment factor Fa for each material that we introduce in order to modify the coefficient ðdE=dXÞE0 r of Eq. (6). This factor Fa modifies the stopping power to a new value given by the expression K ¼ Fa ðdE=dXÞE0 r;
ð7Þ
which is replaced in Eq. (6) to give X ¼ DE=K;
ð8Þ
where K is the new averaged value of the attenuation coefficient. The calculated values using Eq. (8) are shown with a continuous line in Figs. 5(a–f), for different materials, with the adjustment factors (Fa ) listed in Table 1. In the titanium case, used as target, the adjustment factor for this material is obtained in terms of
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184
Fig. 5. Curves for the maximum penetration of the electron beam, at different energies and different materials. The dotted curve is the experimental value, and the continuous line was calculated with the adjustment factor Fa : (a) Air, (b) water, (c) steel, (d) glass, (e) copper, (f) aluminum.
Table 1 Adjustment factors for different materials for energy range from 0.1 to 1.0 Mev Material
Titanium
Air
Water
Steel
Glass
Copper
Aluminum
Fa
1.11
1.62
1.40
2.05
1.64
2.4
1.75
the data reported by Becker et al. (1979). Fig. 6 shows the behavior of the penetration depth of the electrons with respect to the bombarding energy on titanium, the curve was considered as it was suggested.
4. Calculation of the absorbed dose When an electron beam collides with a homogeneous material, the material absorbs a part of the energy of the beam reducing the electrons velocity. The amount of
! G. Vazquez-Polo et al. / Radiation Physics and Chemistry 64 (2002) 181–187
Fig. 6. Curve for the maximum penetration of the electron beam in titanium with adjustment factor Fa :
energy absorbed by a unit of mass is defined as Dose ¼ D ¼ DE=M; where DE is the average absorbed energy from the electron radiation by the matter in (MeV) and DM the irradiated material mass in (g). If Am is the irradiated area of the sample in cm2 and X the electron penetration depth given by Eq. (8), the irradiated material is M ¼ rAm DE=K:
ð9Þ
The energy absorbed by the material is a function of the dose and becomes larger with the penetration of the electrons inside the material. It has a maximum inside the sample and quickly decreases to zero. The depth–dose curve has been calculated by Tanaka et al. (1989); Kobetich and Katz (1968); and by Tabata et al. (1990). They have proposed simple relationships to evaluate the depth–dose for standard geometry. Theoretical calculations of the depth–dose curve are difficult, although the primary interactions of electrons with matter are well known. In order to improve the results, we follow the phenomenological description given by Brynjolfsson (1963). Often, the problem is where to make approximations and what to emphasize. In this attempt, we suggest an empirical expression, which considers the adjustment factor Fa (see Section 3) modulated by an exponential envelope, which, at the same time, is a function of the total penetration of the electrons. This expression is of the form i2 h T CItEi Ei XX XT D¼ 1 e ; ð10Þ DM E0 where D is the total dose in MeV/g, C the constant of unit conversion, Ei the electron’s energy on the surface of the irradiated material in MeV, I the current density as defined in Eq. (3) in mA/cm2, t the irradiation time in s, DM the total mass irradiated defined by Eq. (8) in g: E0 the electron’s initial energy in MeV, X the electron penetration using the adjustment factor given by the
185
Eq. (7) in cm, and XT the electron penetration without taking into account adjustment factor in cm. The suggested empirical expression for the depth–dose curve essentially considers: (a) stopping power (b) nuclear scattering (c) straggling. (a) The analysis of the stopping of the electrons was made by Bethe (1934). It is often said that the fast secondary electrons are the main cause of the dose increase with depth near the surface. In fact, this is not correct, because few fast secondary electrons have energy above 0.2 of the initial electron energy, and then secondary electrons at lower energy become very diffuse and can be locally absorbed as long as the sample and irradiation geometries are large. (b) For low-energy electrons, the nuclear scattering is the effect that mainly determines the form of the depth– dose curve and is approximately proportional to relation dZ=E 2 ; where d is the thickness in g/cm2, Z the atomic number, E the electron energy in MeV. The slope of the curve near the surface is proportional to Z 2 =ðFEÞ2 ; where F is an appropriate factor function of electron energy, Yang (1951). At lower values of Z2 =ðFEÞ2 the surface dose is mainly determined by the stopping power. (c) At the beginning of the curve, where the dose decreases from maximum to zero, the decrease at low Z 2 =ðFEÞ2 values is mainly determined by the straggling while at higher Z2 =ðFEÞ2 values is determined by diffusion. Then, the straggling of the electrons is the main effect determining this slope.
5. Calculation and comparison for some materials Fig. 7 shows the values of D given by Eq. (10) for water for different thickness with electron energy of 0.4 and 0.7 MeV. A comparison with experimental data
Fig. 7. Dose distribution in water as a function of depth, for different energies the dotted line shows the experimental value. The continuous line shows the calculated value.
186
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Fig. 8. Depth–dose distribution curves in polystyrene. The dotted lines show experimental values. Continuous lines show values already calculated by the model at energies of: (a) 400, (b) 500, (c) 600, and (d) 700 keV.
from Becker et al. (1979) shows a good agreement in maximum depth and dose in both curves (shown by continuous line) A and B. Expressions (7) and (9), proposed in this work, were also applied to the irradiation case of polystyrene, with an adjustment factor (Fa ) of 1.2. The depth–dose curves are shown in Figs. 8(a–d) for electron energies of 400, 500, 600 and 700 keV. These curves are compared with the experimental values reported by the ASTM (1993); the calculated values are shown by a continuous line and the experimental data are given by dots. It can be seen that the best fit is obtained for energies larger than 400 keV. Our suggestion was also applied to the case of the radiochromic liquid solution dosimetry system. Fig. 9 shows the result for electron energy of 0.7 MeV in continuous line and the experimental values, reported by Zuazua and Hern!andez (1993), are shown with dots. It shows good agreement between Eq. (9) and the experimental results.
6. Discussion In the characterization of the electron beam, it is worthwhile to mention some feature of the method
Fig. 9. Dose distribution in radio chromic liquid solution dosimetry system irradiated with incident electron energy E0 ¼ 0:7 MeV. The dotted curve shows experimental values and the continuous line shows values calculated by the proposed relationships.
presented here. Firstly, it is very simple and it allows us to have an average value of the current density. However, from the digitized spots of the beam of our system it is possible to infer the distribution profile and
! G. Vazquez-Polo et al. / Radiation Physics and Chemistry 64 (2002) 181–187
we can say that it behaves symmetrically with regard to the center of the beam, which enhances the method of determining Am very precisely to use it in expression (8) which in turn is applied to evaluate Eq. (9), the depth– dose curve. On the other hand, concerning the adjustment factor Fa to evaluate the depth–dose curve, physically it represents the deviation of the processes from the approximation presented here, thus this parameter is phenomenological and it could not be related with first principles, although it adjusts the experimental values in very useful way.
7. Conclusions Comparisons between our results and the experimental values of the depth–dose curve, allow us to conclude that: (A) To calibrate the profile and dimensions of the electron beam of an accelerator, we have suggested a reasonable method that gives an acceptable value of its density current. (B) Considering the mass stopping value times the adjustment factor for each material and experimental values, allows in determining the penetration depth as a function of the incident energy in an easy way. (C) The empirical expression proposed for calculating the depth–dose curve provides good agreement with experimental measurements for energies larger than 0.4 MeV in the case of water, polyester and the radiochromic liquid. However, for energies lower than 0.4 MeV, the prediction values of depth and dose differ slightly from the experimental, but they give us a good insight into the physical phenomenon.
References ASTM, 1993. Technical Committee. Practice for dosimetry in an electron beam facility for radiation processing
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at energies between 300 keV and 25 MeV, E10, 01 appendix 11. Becker, R.C., Bly, J.H., Cleland, M.R., Farrell, J.P., 1979. Accelerator requirements for electron beams processing. Advances in radiation processing. Radiat. Phys. Chem. 14, 353–375. Berger, M.J., Seltzer, S.M., 1982. Stopping Power and Ranges of Electrons and Positrons, 2nd Edition. US Department of Commerce, National Bureau of Standards, Washington, DC, 82-2550-A. Bethe, H.A., 1934. The influence of screening on the creation and stopping of electrons. Proc. Cambridge Philos. Soc. 30, 524–539. Brynjolfsson, A., 1963. Three-dimensional dose distribution in samples irradiated by electron beams. Proceedings of an International Conference, January 14–16, Radiation Research PB 181506, pp. 116–129. Kobetich, E.J., Katz, R., 1968. Energy deposition by electron beams and d rays. Phys. Rev. 170, 391–396. McLaughlin, W.L., Hjortenberg, P.E., Pederson, W.B., 1975. Low energy scanned electron-beam dose distributions in thin layers. Int. J. Appl. Rad. Isot. 2b, 95–106. Nelms, A.T., 1956. Energy loss and range of electrons and positions. National Bureau of Standards circular C-557. Okabe, S., Tsumori, K., Tabata, T., Yoshida, T., Najoi, T., Hiro, S., Hishida, K., Sakumoto, Y., Kawai, T., Arakawa, K., Ivone, T., Murakami, T., 1974. Estimating and measuring methods for absorbed dose of electron. Case of 300 keV accelerators. Oyo Butsuri 43, 909–918. Tabata, T., Ito, R., Kuriyama, I., Moriuchi, Y., 1989. Simple method of evaluation absorbed dose in electron-beam processing. Radiat. Phys. Chem. 33, 411–416. Tabata, T., Ito, R., Tsukui, S., 1990. Semiempirical algorithm for dose evaluation in electron beam processing. Radiat. Phys. Chem. 35, 821–825. Tanaka, R., Sunaga, H., Kuriyama, I., Moriuchi, Y., 1989. Intercomparative study on low energy electron beam dosimetry, radiat. Phys. Chem. 35, 407–410. Yang, C.N., 1951. Actual path length of electrons in foils. Phys. Rev. 84, 599–600. ! Zuazua Garc!ıa, M.P., Hern!andez, M.V., 1993. Caracterizacion del haz de electrones del acelerador Pelletron. IV Congreso Anual de la Sociedad Nuclear Mexicana,Vol. I, pp. 265–270.
Calculation of absorbed dose of low-energy electron beam by an approximate method ! G. Va! zquez-Poloa,*, H. Lopez Valdiviab, H. Carrasco Abregob, R.R. Mijangosc, R. Garc!ıa Ga ! Instituto de F!ısica, UNAM, AP 20-364, Mexico, DF 01000, Mexico ! Instituto Nacional de Investigaciones Nucleares, AP 18-1027, Mexico, DF 11801, Mexico c ! en F!ısica UNISON, AP 5-88.83190, Hermosillo, Son. Mexico Centro de Investigacion a
b
Received 6 April 2000; accepted 4 July 2001
Abstract In this work, we present a simple method to calculate absorbed doses of low-energy electron beams together with an experimental procedure for calibrating the accelerator under certain work conditions. The semi-empirical method is useful for a rapid estimation of depth–dose curve of electron beams, which uses an adjustable parameter factor determined by the least-squares fit from experimental values. Our approach is tested in electron processes on different materials, obtaining reasonable agreement. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Radiation processing; Electron beam; Depth–dose curve; Algorithm
1. Introduction Electron accelerators have a wide range of applications in the industry and in the basic research. Two of the most common problems during material irradiation with electron beams are determining the absorbed dose and the reached depth. The approach in each case depends on the kind of irradiator, the technological resources and the infrastructure available. See McLaughlin et al. (1975) for a review of this topic. Most of the theoretical work for evaluating the absorbed dose in an irradiated material uses computational methods. Monte Carlo simulation and direct numerical solution of transport equation are some useful methods. However, both of them require large CPU time and make necessary the use of high capacity computers. In this work, we present a semi-empirical method using a standard personal computer for *Corresponding author. Fax: +52-5-622-5011. E-mail address: [email protected]fisicacu.unam.mx (G. V!azquez-Polo).
determining points of the depth–dose curve of monoenergetic electrons with an energy range of 0.1–1.0 MeV. The obtained values agree within an acceptable range, with the experimental measurements. This agreement allows us to have a general experimental process to calibrate low-energy electron accelerators. We are making use of this process in the pelletron-type accelerator at the Instituto Nacional de Investigaciones Nucleares (ININ). In irradiation processing, by employing a beam of low-energy electrons, the geometry of the system can be considerably simplified, following the suggestion of Okabe et al. (1974) and Tabata et al. (1989), where essentially a normally incident beam of electrons collides with a three-layer slab absorber: 1. the accelerator window (titanium foil), 2. the air layer and 3. the sample layer. This geometry is an applicable arrangement for a large majority of electron accelerators of the scanningbeam type where the maximum angle of scanning is small.
0969-806X/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 9 - 8 0 6 X ( 0 1 ) 0 0 4 5 9 - 5
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182
2. Electron beam characterization To adequately calibrate an accelerator, it is necessary to know the profile and dimensions of the electron beam, besides the current intensity. These parameters can be evaluated by the following procedure. Two plates of PVC are exposed to the electron beam in order to have a permanent and visible mark in the sample. These plates are placed, one on site A; at a distance XA from the titanium foil window. The other one is placed in point B covering the aluminum curtain at a distance XB ; as shown in Fig. 1. The marks over the plates produced by the electron beam can be digitized in order to have a better knowledge of the real shape and dimensions of the electron beam in this region. In Fig. 2, we can see the digitized spots of the PVC plates and the spots showing their real size are also visible in Figs. 3a and b, respectively, from which an effective spot radius rA and rB can be evaluated. Using these measurements, it is possible to calculate the effective density of the current and the maximum irradiation area in the sample. For these goals, a virtual point focus located in F at a distance d from the titanium window, is geometrically defined as shown in Fig. 4 (the thickness of the titanium windows is most likely about 40 mm) This distance due to the geometry involved is given by d¼
X B rA X A rB : rB rA
ð1Þ
If the sample is located at distance Xm from the titanium window, the radius of the area is given by rM ¼
rB ðd þ Xm Þ d þ XB
ð2Þ
and the area Am covered by the electron beam over the sample can be calculated. The density of the current of the electron beam that strikes the sample, supposing that
Fig. 2. Digitized spots of the marks over the PVC plates produced by the electron beam.
the beam is homogeneous and monoenergetic is given by I¼
I0 ; Am
ð3Þ
where I0 is the current, in microamperes, measured on the aluminum screen and Am the irradiated area in the sample, given in cm2.
3. Adjustment factor for energy-depth determination In order to evaluate the energy loss of the electron beam when it passes through a given material of thickness DX; the following general expression can be used: DE ¼ E0 Ef ¼ ðdE=dXÞE0 r DX;
ð4Þ
where DE is the kinetic energy lost by the electron beam (MeV), E0 the kinetic energy of the incident electron, before hitting the titanium window (Mev) in this case the profile is 2.4 2.8 cm2 of area, Ef the kinetic energy of the transmitted electron (MeV), ðdE=dXÞE0 the stopping power of the sample for electrons with initial energy E0 (MeV cm2/g), r the density of the material (g/cm3), and DX the thickness in cm penetrated by the electron. From Eq. (4), we get
Fig. 1. View of the electron beam accelerator showing the position of: focus (F), scanner’s windows (V), PVC plates (A, B) and sample (C).
DE DX ¼ : ðdE=dXÞE0 r
ð5Þ
From the experimental data of Nelms (1956), it can be seen that the incremental depth penetration in the
! G. Vazquez-Polo et al. / Radiation Physics and Chemistry 64 (2002) 181–187
183
Fig. 4. Virtual focus point (F) created geometrically.
Fig. 3. Digitized spots of the PVC plates in A and B positions (Fig. 1), respectively.
considered energy range is proportional to the electron’s energy loss DE: Now we divide the penetrated sample into DXn segments using for each one ðdE=dXÞn shown in the tables given by Berger and Seltzer (1982) for the corresponding energy DEn of the segments, where DEn is the fraction of the incident energy E0 ; then we can sum these thickness segments up to energy E0 and we then get X¼
nX ¼N n¼1
DXn ¼
X
DEn ðdE=dxÞn
r:
ð6Þ
Eq. (6) can be evaluated for different energies, in the range 0.1 to 1.0 MeV plot X ¼ XE0 as a function of E0 using the electron stopping power values of the National
Bureau of Standards reported by Berger and Seltzer (1982) which take into account: (a) the ionization and excitation losses (collision stopping power) and (b) the bremsstrahlung losses (radiation stopping powers). In a strict sense, this is, still, only an approximation to the theoretical predictions, because we have assumed that the energy of the electrons inside each thickness is homogenous and the exit energy is Ef ¼ E0 DE; as it was assumed in Eq. (4). Additionally, for each value of electron energy inside the material, we must consider the corresponding stopping power value. Taking into account this fact, we modify expression (5) by replacing the interaction between the electron beam and the target with a parameter, which considers the average values of the stopping power factor inside the materials as follows. We propose to adjust the experimental values taken from Nelms (1956), by means of least-square fitting, and relating their slope with the approximated values which gives an adjustment factor Fa for each material that we introduce in order to modify the coefficient ðdE=dXÞE0 r of Eq. (6). This factor Fa modifies the stopping power to a new value given by the expression K ¼ Fa ðdE=dXÞE0 r;
ð7Þ
which is replaced in Eq. (6) to give X ¼ DE=K;
ð8Þ
where K is the new averaged value of the attenuation coefficient. The calculated values using Eq. (8) are shown with a continuous line in Figs. 5(a–f), for different materials, with the adjustment factors (Fa ) listed in Table 1. In the titanium case, used as target, the adjustment factor for this material is obtained in terms of
! G. Vazquez-Polo et al. / Radiation Physics and Chemistry 64 (2002) 181–187
184
Fig. 5. Curves for the maximum penetration of the electron beam, at different energies and different materials. The dotted curve is the experimental value, and the continuous line was calculated with the adjustment factor Fa : (a) Air, (b) water, (c) steel, (d) glass, (e) copper, (f) aluminum.
Table 1 Adjustment factors for different materials for energy range from 0.1 to 1.0 Mev Material
Titanium
Air
Water
Steel
Glass
Copper
Aluminum
Fa
1.11
1.62
1.40
2.05
1.64
2.4
1.75
the data reported by Becker et al. (1979). Fig. 6 shows the behavior of the penetration depth of the electrons with respect to the bombarding energy on titanium, the curve was considered as it was suggested.
4. Calculation of the absorbed dose When an electron beam collides with a homogeneous material, the material absorbs a part of the energy of the beam reducing the electrons velocity. The amount of
! G. Vazquez-Polo et al. / Radiation Physics and Chemistry 64 (2002) 181–187
Fig. 6. Curve for the maximum penetration of the electron beam in titanium with adjustment factor Fa :
energy absorbed by a unit of mass is defined as Dose ¼ D ¼ DE=M; where DE is the average absorbed energy from the electron radiation by the matter in (MeV) and DM the irradiated material mass in (g). If Am is the irradiated area of the sample in cm2 and X the electron penetration depth given by Eq. (8), the irradiated material is M ¼ rAm DE=K:
ð9Þ
The energy absorbed by the material is a function of the dose and becomes larger with the penetration of the electrons inside the material. It has a maximum inside the sample and quickly decreases to zero. The depth–dose curve has been calculated by Tanaka et al. (1989); Kobetich and Katz (1968); and by Tabata et al. (1990). They have proposed simple relationships to evaluate the depth–dose for standard geometry. Theoretical calculations of the depth–dose curve are difficult, although the primary interactions of electrons with matter are well known. In order to improve the results, we follow the phenomenological description given by Brynjolfsson (1963). Often, the problem is where to make approximations and what to emphasize. In this attempt, we suggest an empirical expression, which considers the adjustment factor Fa (see Section 3) modulated by an exponential envelope, which, at the same time, is a function of the total penetration of the electrons. This expression is of the form i2 h T CItEi Ei XX XT D¼ 1 e ; ð10Þ DM E0 where D is the total dose in MeV/g, C the constant of unit conversion, Ei the electron’s energy on the surface of the irradiated material in MeV, I the current density as defined in Eq. (3) in mA/cm2, t the irradiation time in s, DM the total mass irradiated defined by Eq. (8) in g: E0 the electron’s initial energy in MeV, X the electron penetration using the adjustment factor given by the
185
Eq. (7) in cm, and XT the electron penetration without taking into account adjustment factor in cm. The suggested empirical expression for the depth–dose curve essentially considers: (a) stopping power (b) nuclear scattering (c) straggling. (a) The analysis of the stopping of the electrons was made by Bethe (1934). It is often said that the fast secondary electrons are the main cause of the dose increase with depth near the surface. In fact, this is not correct, because few fast secondary electrons have energy above 0.2 of the initial electron energy, and then secondary electrons at lower energy become very diffuse and can be locally absorbed as long as the sample and irradiation geometries are large. (b) For low-energy electrons, the nuclear scattering is the effect that mainly determines the form of the depth– dose curve and is approximately proportional to relation dZ=E 2 ; where d is the thickness in g/cm2, Z the atomic number, E the electron energy in MeV. The slope of the curve near the surface is proportional to Z 2 =ðFEÞ2 ; where F is an appropriate factor function of electron energy, Yang (1951). At lower values of Z2 =ðFEÞ2 the surface dose is mainly determined by the stopping power. (c) At the beginning of the curve, where the dose decreases from maximum to zero, the decrease at low Z 2 =ðFEÞ2 values is mainly determined by the straggling while at higher Z2 =ðFEÞ2 values is determined by diffusion. Then, the straggling of the electrons is the main effect determining this slope.
5. Calculation and comparison for some materials Fig. 7 shows the values of D given by Eq. (10) for water for different thickness with electron energy of 0.4 and 0.7 MeV. A comparison with experimental data
Fig. 7. Dose distribution in water as a function of depth, for different energies the dotted line shows the experimental value. The continuous line shows the calculated value.
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Fig. 8. Depth–dose distribution curves in polystyrene. The dotted lines show experimental values. Continuous lines show values already calculated by the model at energies of: (a) 400, (b) 500, (c) 600, and (d) 700 keV.
from Becker et al. (1979) shows a good agreement in maximum depth and dose in both curves (shown by continuous line) A and B. Expressions (7) and (9), proposed in this work, were also applied to the irradiation case of polystyrene, with an adjustment factor (Fa ) of 1.2. The depth–dose curves are shown in Figs. 8(a–d) for electron energies of 400, 500, 600 and 700 keV. These curves are compared with the experimental values reported by the ASTM (1993); the calculated values are shown by a continuous line and the experimental data are given by dots. It can be seen that the best fit is obtained for energies larger than 400 keV. Our suggestion was also applied to the case of the radiochromic liquid solution dosimetry system. Fig. 9 shows the result for electron energy of 0.7 MeV in continuous line and the experimental values, reported by Zuazua and Hern!andez (1993), are shown with dots. It shows good agreement between Eq. (9) and the experimental results.
6. Discussion In the characterization of the electron beam, it is worthwhile to mention some feature of the method
Fig. 9. Dose distribution in radio chromic liquid solution dosimetry system irradiated with incident electron energy E0 ¼ 0:7 MeV. The dotted curve shows experimental values and the continuous line shows values calculated by the proposed relationships.
presented here. Firstly, it is very simple and it allows us to have an average value of the current density. However, from the digitized spots of the beam of our system it is possible to infer the distribution profile and
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we can say that it behaves symmetrically with regard to the center of the beam, which enhances the method of determining Am very precisely to use it in expression (8) which in turn is applied to evaluate Eq. (9), the depth– dose curve. On the other hand, concerning the adjustment factor Fa to evaluate the depth–dose curve, physically it represents the deviation of the processes from the approximation presented here, thus this parameter is phenomenological and it could not be related with first principles, although it adjusts the experimental values in very useful way.
7. Conclusions Comparisons between our results and the experimental values of the depth–dose curve, allow us to conclude that: (A) To calibrate the profile and dimensions of the electron beam of an accelerator, we have suggested a reasonable method that gives an acceptable value of its density current. (B) Considering the mass stopping value times the adjustment factor for each material and experimental values, allows in determining the penetration depth as a function of the incident energy in an easy way. (C) The empirical expression proposed for calculating the depth–dose curve provides good agreement with experimental measurements for energies larger than 0.4 MeV in the case of water, polyester and the radiochromic liquid. However, for energies lower than 0.4 MeV, the prediction values of depth and dose differ slightly from the experimental, but they give us a good insight into the physical phenomenon.
References ASTM, 1993. Technical Committee. Practice for dosimetry in an electron beam facility for radiation processing
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at energies between 300 keV and 25 MeV, E10, 01 appendix 11. Becker, R.C., Bly, J.H., Cleland, M.R., Farrell, J.P., 1979. Accelerator requirements for electron beams processing. Advances in radiation processing. Radiat. Phys. Chem. 14, 353–375. Berger, M.J., Seltzer, S.M., 1982. Stopping Power and Ranges of Electrons and Positrons, 2nd Edition. US Department of Commerce, National Bureau of Standards, Washington, DC, 82-2550-A. Bethe, H.A., 1934. The influence of screening on the creation and stopping of electrons. Proc. Cambridge Philos. Soc. 30, 524–539. Brynjolfsson, A., 1963. Three-dimensional dose distribution in samples irradiated by electron beams. Proceedings of an International Conference, January 14–16, Radiation Research PB 181506, pp. 116–129. Kobetich, E.J., Katz, R., 1968. Energy deposition by electron beams and d rays. Phys. Rev. 170, 391–396. McLaughlin, W.L., Hjortenberg, P.E., Pederson, W.B., 1975. Low energy scanned electron-beam dose distributions in thin layers. Int. J. Appl. Rad. Isot. 2b, 95–106. Nelms, A.T., 1956. Energy loss and range of electrons and positions. National Bureau of Standards circular C-557. Okabe, S., Tsumori, K., Tabata, T., Yoshida, T., Najoi, T., Hiro, S., Hishida, K., Sakumoto, Y., Kawai, T., Arakawa, K., Ivone, T., Murakami, T., 1974. Estimating and measuring methods for absorbed dose of electron. Case of 300 keV accelerators. Oyo Butsuri 43, 909–918. Tabata, T., Ito, R., Kuriyama, I., Moriuchi, Y., 1989. Simple method of evaluation absorbed dose in electron-beam processing. Radiat. Phys. Chem. 33, 411–416. Tabata, T., Ito, R., Tsukui, S., 1990. Semiempirical algorithm for dose evaluation in electron beam processing. Radiat. Phys. Chem. 35, 821–825. Tanaka, R., Sunaga, H., Kuriyama, I., Moriuchi, Y., 1989. Intercomparative study on low energy electron beam dosimetry, radiat. Phys. Chem. 35, 407–410. Yang, C.N., 1951. Actual path length of electrons in foils. Phys. Rev. 84, 599–600. ! Zuazua Garc!ıa, M.P., Hern!andez, M.V., 1993. Caracterizacion del haz de electrones del acelerador Pelletron. IV Congreso Anual de la Sociedad Nuclear Mexicana,Vol. I, pp. 265–270.