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Comparative modelling of X-ray backscattering in the Novillo tokamak
Radiation Measurements 33 (2001) 409–416
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Comparative modelling of X-ray backscattering in the Novillo tokamak A. Floresa; 1 , A. Castilloa , S.R. Barocioa , E. Ch/aveza , L. Mel/endeza , G.J. Cruza , M.G. Olayoa , P. Gonz/aleza , J. Azor/6nb; c; ∗ a Instituto
Nacional de Investigaciones Nucleares, 52045 Salazar, Mexico Aut onoma Metropolitana-Iztapalapa, 09340 M exico, D.F. Mexico c Centro de Investigaci on en Ciencia Aplicada y Tecnolog (a Avanzada-IPN, 11500 M exico, D.F. Mexico b Universidad
Received 3 March 2000; received in revised form 7 December 2000; accepted 24 December 2000
Abstract Results of X-ray backscattering studies on the Novillo tokamak using thermoluminescent dosimeters (TLDs) are presented. Dose measurements were carried out on the equatorial plane of the tokamak along 12 radial directions. The backscattering can be due to the interaction of the radiation with the surrounding walls. Two kinds of phenomenological mathematical methods describing the X-ray backscattering were obtained from the experimental data and compared. Good agreement was found c 2001 Published by Elsevier Science Ltd. between the data and the predictions of the models.
1. Introduction X-ray detection by means of thermoluminescent dosimeters (TLDs) has proved to be a useful diagnostic technique for tokamaks (Tanahashi, 1981; Ramos et al., 1983). We have carried out thermoluminescent measurements of the X-ray emission during discharges previously on the Novillo tokamak (Flores et al., 1998). As for other tokamaks, Youhua et al. (1988) have reported that the variation in dose is inversely proportional to the distance from the tokamak vessel and that its spatial distribution is directionally anisotropic. Yin-Xiejin et al. (1988) have shown that the X-ray proAle has a continuous spectrum whose intensity depends on distance and direction. It has also been reported from the Fontenay-aux-Roses tokamak (TFR) that the normalised X-ray reCection patterns created by the surrounding walls can reveal several characteristics of its electron beam source (CEN, 1979). Other authors (Tanahashi, 1981) have estimated the dose both on non-shielded locations close ∗ Corresponding author. Centro de Investigaci/ on en Ciencia Aplicada y Tecnolog/6a Avanzada-IPN, 11500 M/exico, D.F. Mexico. Tel.: +52-5804-4620; fax: +52-5804-4611. E-mail address: [email protected] (J. Azor/6n). 1 In memoriam.
to a hybrid tokamak-stellerator machine and outside the building housing it, using TLDs. In this work, two heuristic mathematical models for the X-ray emission from runaway electrons in tokamak plasma are proposed. The models are developed out of the experimental determination of X-ray dose around the Novillo tokamak as a function of the radial distance to its vessel wall and of the plasma current IP ; using TLDs. A good agreement was found between the experimental data and the proposed models. Their respective advantages and limitations are comparatively presented. 2. Experimental set-up The TLD detectors were placed on the tokamak equatorial plane according to the following distribution: 12 radial rows at every 30◦ toroidal angle, each row containing four equidistant detectors, 0.5 m apart from each other. Previous to every plasma discharge, the background radiation level was established from the dose measurements. The TLDs used to take the dose measurements were especially manufactured 0.8 mm thick CaSO4 : Dy + PTFE pellets, 5 mm in diameter (Azor/6n and Guti/errez, 1989), due to its sharp response in the 0.5 –1.5 MeV energy range, bearing in mind
c 2001 Published by Elsevier Science Ltd. 1350-4487/01/$ - see front matter PII: S 1 3 5 0 - 4 4 8 7 ( 0 1 ) 0 0 0 3 2 - 4
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A. Flores et al. / Radiation Measurements 33 (2001) 409–416
that the eJective energy of the X-rays from the tokamak is ∼1:0 MeV (Mirnov, 1998). In order to produce reliable data about the eJective tokamak energy emission, the detectors were calibrated using a 137 Cs source standard so that, above the threshold interval 30 –150 kV, the response of the TLDs were linear and independent of the absorbed energy. The angular position of the limiter, namely, the metallic diaphragm that deAnes the border of the plasma column inside the vessel, has been taken as a reference point to set the origin of the angular displacement ( = 0◦ ). The tokamak pre-ionisation Alament is located at an angle = 300◦ between the TLDs in rows 10 and 11. The experiment was performed during two series of 30 tokamak discharges. The Arst series with a plateau level plasma current IP =4 kA, and the second one with IP =8 kA. The duration of each discharge was approximately 600 s.
3. Results and discussion Fig. 1 shows some representative samples of the general behaviour of X-ray emission as measured from and around the tokamak vessel wall. We adopted gray (Gy) as the only
Fig. 2. Typical X-ray dose plotted as a function of the radial and angular distribution of detectors. (a) Plasma current IP = 4000 A, (b) Plasma current IP = 8000 A.
Fig. 1. Absorbed dose (Gy) as a function of the distance from the tokamak vessel wall (m) for two levels of plasma current (4000 A, 8000 A), at the following toroidal angle directions: (a) = 0◦ , (b) = 210◦ , (c) = 330◦ .
radiation unit in this work because CaSO4 : Dy TLD detectors register the (time-integrated) dose of energy absorbed by mass unit. So, the vertical axis of the plots refers to the absorbed dose in Gy while the horizontal axis to the distance from the wall in m. The presence of maxima in the proximity of the vessel for both plasma current levels can be noticed. Further away, the curves fall to a minimum just to take a gradual ascent and Anally decay. As a starting point of the model development, all data were plotted in cylinder coordinate coaxial to the tokamak, detecting the common characteristics of their behaviour. To simplify our subsequent data processing and plotting, a Cartesian coordinate system was adopted later, where x = r cos , y = r sin and the z-axis was reserved for the absorbed dose. Figs. 2(a) and 2(b) exhibit, in an r– slab geometry for IP = 4 kA and 8 kA cases, respectively, the data set chosen to establish the models, whose points have been linked together so to help identify their behaviour. Once working in the x–y Cartesian coordinate system, two main procedures were followed. For a start, the
A. Flores et al. / Radiation Measurements 33 (2001) 409–416
experimental data were Atted through a least squares algorithm into a bicubic polynomial. Then, a more sophisticated mathematical model was examined so as to approach a physical interpretation of the phenomenon in the best possible way, leading to the completion of the models proposed in this work. The choice of a bicubic polynomial was made due to the overall radial characteristics of all the analysed dose proAles, namely: the presence of a maximum, a minimum and an asymptotic decay, to be reCected in the radial dependence of x and y variables. So, the next step was to propose a Taylor-like polynomial in the following form: P(x; y) = a1 + a2 x + a3 x2 + a4 x3 + a5 y + a6 xy + a7 x2 y + a8 x3 y + a9 y2 + a10 xy2 + a11 x2 y2 + a12 x3 y2 + a13 y3 + a14 xy3 + a15 x2 y3 + a16 x3 y3 ; where x = r cos , y = r sin . The polynomial P(x; y) represents the dose due to the radiation emitted by the tokamak. Clearly, the dose density constants ai must have dimensions according to the power of the variables they multiply. From a more physical point of view, P(x; y) can be interpreted as the superimposed contribution of several harmonics or equivalent sources of radiation, each one of them with a definite spatial Cuctuation form, and a diJerent speciAc weight ai , within the experimental conAnes. Such harmonic contributions model the rather complex distribution topology of X-ray emission, reCection and absorption, interacting within and around the tokamak vessel. In order to determine the coeLcients ai , experimental data were incorporated as constraints in the form of an overdetermined linear system A · x = b; where x = x(ai ) is a vectorial array, i = 1; 2; : : : ; 16. The results presented in Table 1 were obtained by solving this system by least squares. Rounding these results, in each case of tokamak plasma current considered, the polynomial appears as: P4 (x; y) = (25:2 Gy) + (1 Gy m−1 )x + (28:1 Gy m−1 )y − (2:7 Gy m−2 )xy − (16:9 Gy m−2 )y2 + (2:2 Gy m−3 )xy2 + (2:9 Gy m−3 )y3 − (0:5 Gy m−4 )xy3 ; P8 (x; y) = (11:3 Gy) + (1:6 Gy m−1 )x + (44:2 Gy m−1 )y − (3:8 Gy m−2 )xy − (52:1 Gy m−2 )y2 + (4:4 Gy m−3 )xy2 + (17:2 Gy m−3 )y3 − (1:4 Gy m−4 )xy3 ; the subindex on the left-hand side of the equations indicates the respective value of the current in kA. It can be observed that, within the limits of the collected data volume and with the exception of two terms, an amplitude ratio of approximately two exists between both polynomials. It is signiAcant to observe that P(x; y) is far from being isotropic and that its strongest variations occur in the
411
Table 1 Adjusted values of the polynomial coeLcients for two levels of tokamak plasma current. The coeLcients are presented in a dimensionless form for the sake of clarity CoeLcient a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16
4 kA
8 kA 10−2
2:523 × 1:0167 × 10−3 −6:86 × 10−6 1:0 × 10−8 2:8114 × 10−2 −2:7138 × 10−3 1:784 × 10−5 −2:0 × 10−8 −1:6931 × 10−2 2:1729 × 10−3 −1:455 × 10−5 2:0 × 10−8 2:9353 × 10−3 −5:2039 × 10−4 3:53 × 10−6 −5:0 × 10−9
1:1257 × 10−2 1:5750 × 10−3 −4:07 × 10−6 −5:0 × 10−7 4:4188 × 10−2 −3:8459 × 10−3 1:36 × 10−5 7:0 × 10−8 −5:2105 × 10−2 4:409 × 10−3 −2:157 × 10−5 1:0 × 10−8 1:7182 × 10−2 −1:383 × 10−3 7:7 × 10−6 −8:0 × 10−9
y-direction, mainly with linear and quadratic dependencies. This fact can be attributed to the toroidal location of the tokamak plasma limiter, whose purpose is to shear the plasma edge to a speciAc minor radius. Thus, it is highly probable that the edge plasma electrons colliding against the limiter surface emit bremsstrahlung radiation. P(x; y) behaviour is shown in Figs. 3 and 4 In them, the obtained polynomials are plotted both on x–y and r– slab meshes Aner than the original (experimental) one. Such dual geometry will be maintained in all the subsequent Agures for the sake of a clearer interpretation and every data point will be linked to its respective location predicted by the model, in order to assess the accuracy of it. Searching for an alternative, more analytical, data Atting relying on a smaller parameter set, particular attention was paid to: (a) The experimental behaviour in the 0 –2 m distance domain where the available data were more equidistantly recorded. (b) The data distribution throughout the sampled distance, especially far away from the tokamak. (c) The cross-sectional topology of the Atting at constant angles. So, it was considered that the steepest changes in the dose distribution occurred within a distance of about 3 m from the machine. Beyond it, the behaviour is assumed to be practically linear with a declining trend. The periodic nature of the close range dose reading led to the choice of sinusoidal functions to be included in the modelling, which should satisfy the observed characteristic of the number of oscillations growing inversely with the size of the plasma current.
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A. Flores et al. / Radiation Measurements 33 (2001) 409–416
Fig. 3. Behaviour of the bicubic polynomial model for the case of a tokamak plasma current IP = 4000 A; (a) in x–y geometry (b) in r– geometry with the angle scaled in radians.
The fundamental sinusoidal models developed, where a, b, c, d and e are Ave suitably dimensioned Atting parameters, x = r cos , y = r sin and F is the absorbed dose in Gy: F(x; y) = [eay sin(by)][c sin(d x)] + e and F(x; y) = ay sin(bx) cos(cy) + d: Figs. 5 and 6 exhibit the behaviour of the Arst of the sinusoidal models, while Figs. 7 and 8 refer to the second one. The optimised values of the function parameters applied in Figs. 5 and 6 are presented in non-dimensional form in Table 2. Physically speaking, in assuming such functional forms we are allowing the problem to be implicitly related to a two dimensional propagation of standing waves, so as to better encompass the interaction between the radiation emission and reCection. At the same time, an attenuation functional
Fig. 4. Behaviour of the bicubic polynomial model for the case of a tokamak plasma current IP = 8000 A; (a) in x–y geometry (b) in r– geometry with the angle scaled in radians.
factor may in principle account for the overall absorption eJects. In order to work out the optimisation of the models, a Levenberg–Marquardt non-linear method was employed. According to which, a second order Taylor expansion of the squared norm of the residuals (sum of the squared diJerences between each data point and its model value) is minimised in the space of the model parameters (Barrera and Olvera, 1993). To that purpose, the so-called Mor/e algorithm (Mor/e and Wright, 1993) was applied to the problem in FORTRAN 77 language. Then, after developing more than ten increasingly reAned versions, the best results were obtained with the following one: √ k4 F(x; y) = k sin x ± 3 y + k k sin(y) y k where k = IP (IP in kA), = 0:3 m Gy1=2 , = 4:525 m−1 , = 0:43 Gy m−1 , = 1:6 Gy, = 0:4 Gy1=2 , = 0:375 Gy−1 and = 0:1 Gy=kA with the upper (+) or
A. Flores et al. / Radiation Measurements 33 (2001) 409–416
Fig. 5. Experimental data Atted according to the Arst sinusoidal model in (a) x–y geometry and (b) r– geometry, with a plasma current IP = 4000 A.
lower (−) signs applying for IP = 4 kA or 8 kA cases, respectively. Figs. 9 and 10 display the data and their predictions according to this Anal model. The functional diJerences between the two factors in square brackets point to the expected anisotropy of the tokamak radiation. As in the case of the polynomial model, the greater part of the Cuctuations takes place along the y-axis. Yet, in the present case, it is clear that the non-oscillatory dependence on y is linear, while the divergence at y = 0 is prevented by the factor sin(y). In both the polynomial and sinusoidal models it is clear that a careful compromise has been reached between the precision of the adjustment and the number of parameters involved. A global estimate of the accuracy of the models is presented in Table 3 on the basis of relative and absolute residuals as well as their norm. To gain a more at glance insight into the Atting quality of the models, Figs. 3–10 exhibit all the residuals in the form of vertical segments.
413
Fig. 6. Experimental data Atted according to the Arst sinusoidal model in (a) x–y geometry and (b) r– geometry, with a plasma current IP = 8000 A.
From them, one can notice a tendency of the large residuals to accumulate at small angles and radii. This fact points to the locally jagged radiation anisotropy created by the tokamak limiter, which is not easy for the models to follow. Not surprisingly, IP = 4 kA plasma regime turns out to be less diLcult to model due to its more even distribution of radiation. On the other hand, the sinusoidal model appears to be slightly less accurate on account of its smaller parameter set. 4. Conclusions The heuristic model processing presented in this work provides a satisfactory Atting of the backscattering phenomenon, to the extent of the data availability. In both the polynomial and sinusoidal models it is clear that a careful compromise has been reached between the precision of the adjustment and the number of parameters involved.
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A. Flores et al. / Radiation Measurements 33 (2001) 409–416
Fig. 7. Experimental data Atted by the second sinusoidal model in (a) x–y and (b) r– geometry, with a plasma current IP = 4000 A.
Fig. 8. Experimental data Atted by the second sinusoidal model in (a) x–y and (b) r– geometry, with a plasma current IP = 8000 A.
Table 2 Values of the Atting parameters employed in Figs. 6 –9. Notice that their dimensions (not shown) may change according to the model Model
Plasma current (kA)
a
b
c
d
e
1
4 8
−1:1137 −0:8427
0.00514 0.003617
−316:71 2968.4
0.00779 0.00628
4.02 3.384
2
4 8
−0:4131 −1:303
5.044 5.0
3.714 6.3
non-existent
Furthermore, the formulation presented here sets the ground for a model evolution leading to an increasing consistency with the involved physical theories. Comparatively speaking, the developed sinusoidal models provide a more applicable structure with respect to the variations of plasma current and with respect to the radial distance beyond the experimental domain while requiring fewer parameters. In fact, these models depend mostly
5.355 5.362
on just one eJective parameter, as all the rest are unitary dimensional coeLcients. However, subsequent models do not necessarily invalidate the previous ones. So, the polynomial variety has proved to be simpler to manipulate provided that its structure guarantees continuous diJerentiability. Polynomials allow greater Cexibility of Atting, as they lend themselves to least square Atting techniques, and the dimensionality of their terms can be balanced without resorting to unitary coeLcients.
A. Flores et al. / Radiation Measurements 33 (2001) 409–416
Fig. 9. Final model Atting the experimental data in (a) x–y and in (b) r– geometry, for the case of a tokamak plasma current IP = 4000 A.
415
Fig. 10. Final model Atting the experimental data in (a) x–y and in (b) r– geometry, for the case of a tokamak plasma current IP = 8000 A.
Table 3 Estimated global precision of the polynomial and sinusoidal models for the IP = 4 kA and 8 kA regimes. The norm of the residuals has been calculated as the square root of the sum of the squared errors Polynomial
Norm of residuals=Gy Absolute error (max) Gy Absolute error (min)=Gy Relative error (max) Relative error (min)
Sinusoidal
IP = 4 kA
IP = 8 kA
IP = 4 kA
IP = 8 kA
0.1939 0.1010 9:0132 × 10−5 0.2025 2:4035 × 10−4
0.524 0.2284 0.0051 0.4117 0.0061
0.2272 0.1251 4:8226 × 10−4 0.2507 0.0013
1.122 0.4175 0.0021 1.7507 0.0034
References Azor/6n, J., Guti/errez, A., 1989. Preparation and performance of a CaSO4 : Dy; Tm thermoluminescent phosphor for long-term gamma measurements. Health Phys. 56, 551–559.
Barrera, P.Y., Olvera, E., 1993. Cuadrados M/6nimos No-lineales. Desarrollos Recientes I. Universidad Nacional Aut/onoma de M/exico, Mexico, D.F. CEN, 1979. Radiation shielding near a controlled fusion research facility. The Fontenay-aux-Roses Tokamak T.F.R.
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Technical Report of the Centre D’Etudes Nuclearies de Fontenay-aux-Roses, 92 Dept. de Physique du Plasma et de la Fusion Controlee, Fontenay-aux-Roses. Flores, A., Mel/endez, L., Ch/avez, E., L/opez, G., Olayo, M.G., Gonz/alez, P., Cruz, G.J., Valencia, R., Rojas, R., Azor/6n, J., S/anchez, A., Sep/ulveda, F., 1998. X-ray thermoluminescent dosimetry around the NOVILLO TOKAMAK. Nucl. Sci. J. 35 (1), 55–59. Mirnov, S., 1998. Report IAEA-RU-7971, Nuclear Technology Support Project (RLA=0=016-6701), International Atomic Energy Agency, Vienna. Mor/e, J.J., Wright, S.J., 1993. Optimization Software Guide, Vol. 14. Society for Industrial and Applied Mathematics, Philadelphia, pp. 15 –19. Ramos, J., de Urquijo, J., Mel/endez, L, Mu˜noz, A., Barocio, S.R., Ch/avez, E., Balderas, E.G., God/6nez, E., Valencia, R., 1983.
Dise˜no del Tokamak Novillo. Rev. Mexicana F/6sica 29 (4), 551–592. Tanahashi, S., 1981. Dose measurement of X-ray radiation from JIPP T-II device in the vicinity of Water Research Institute. Technical Report from Nagoya University, Japan, Institute of Plasma Physics, Nagoya. Yin-Xiejin, Zhang-Huaixian, Liu-Huijun, Wang-Mingjing, Zhen-Lizhen, Li-Xuxion, Lia-Zhengjing, 1988. Study of X-ray dose. Report from China Nuclear Information Centers, Beijing. Youhua, J., Xingying, Z., Taihua, L., 1988. Dose measurements of hard X-ray radiation from HL-1 Tokamak. Nucl. Fusion Plasma Phys. 8, 185–187.
www.elsevier.com/locate/radmeas
Comparative modelling of X-ray backscattering in the Novillo tokamak A. Floresa; 1 , A. Castilloa , S.R. Barocioa , E. Ch/aveza , L. Mel/endeza , G.J. Cruza , M.G. Olayoa , P. Gonz/aleza , J. Azor/6nb; c; ∗ a Instituto
Nacional de Investigaciones Nucleares, 52045 Salazar, Mexico Aut onoma Metropolitana-Iztapalapa, 09340 M exico, D.F. Mexico c Centro de Investigaci on en Ciencia Aplicada y Tecnolog (a Avanzada-IPN, 11500 M exico, D.F. Mexico b Universidad
Received 3 March 2000; received in revised form 7 December 2000; accepted 24 December 2000
Abstract Results of X-ray backscattering studies on the Novillo tokamak using thermoluminescent dosimeters (TLDs) are presented. Dose measurements were carried out on the equatorial plane of the tokamak along 12 radial directions. The backscattering can be due to the interaction of the radiation with the surrounding walls. Two kinds of phenomenological mathematical methods describing the X-ray backscattering were obtained from the experimental data and compared. Good agreement was found c 2001 Published by Elsevier Science Ltd. between the data and the predictions of the models.
1. Introduction X-ray detection by means of thermoluminescent dosimeters (TLDs) has proved to be a useful diagnostic technique for tokamaks (Tanahashi, 1981; Ramos et al., 1983). We have carried out thermoluminescent measurements of the X-ray emission during discharges previously on the Novillo tokamak (Flores et al., 1998). As for other tokamaks, Youhua et al. (1988) have reported that the variation in dose is inversely proportional to the distance from the tokamak vessel and that its spatial distribution is directionally anisotropic. Yin-Xiejin et al. (1988) have shown that the X-ray proAle has a continuous spectrum whose intensity depends on distance and direction. It has also been reported from the Fontenay-aux-Roses tokamak (TFR) that the normalised X-ray reCection patterns created by the surrounding walls can reveal several characteristics of its electron beam source (CEN, 1979). Other authors (Tanahashi, 1981) have estimated the dose both on non-shielded locations close ∗ Corresponding author. Centro de Investigaci/ on en Ciencia Aplicada y Tecnolog/6a Avanzada-IPN, 11500 M/exico, D.F. Mexico. Tel.: +52-5804-4620; fax: +52-5804-4611. E-mail address: [email protected] (J. Azor/6n). 1 In memoriam.
to a hybrid tokamak-stellerator machine and outside the building housing it, using TLDs. In this work, two heuristic mathematical models for the X-ray emission from runaway electrons in tokamak plasma are proposed. The models are developed out of the experimental determination of X-ray dose around the Novillo tokamak as a function of the radial distance to its vessel wall and of the plasma current IP ; using TLDs. A good agreement was found between the experimental data and the proposed models. Their respective advantages and limitations are comparatively presented. 2. Experimental set-up The TLD detectors were placed on the tokamak equatorial plane according to the following distribution: 12 radial rows at every 30◦ toroidal angle, each row containing four equidistant detectors, 0.5 m apart from each other. Previous to every plasma discharge, the background radiation level was established from the dose measurements. The TLDs used to take the dose measurements were especially manufactured 0.8 mm thick CaSO4 : Dy + PTFE pellets, 5 mm in diameter (Azor/6n and Guti/errez, 1989), due to its sharp response in the 0.5 –1.5 MeV energy range, bearing in mind
c 2001 Published by Elsevier Science Ltd. 1350-4487/01/$ - see front matter PII: S 1 3 5 0 - 4 4 8 7 ( 0 1 ) 0 0 0 3 2 - 4
410
A. Flores et al. / Radiation Measurements 33 (2001) 409–416
that the eJective energy of the X-rays from the tokamak is ∼1:0 MeV (Mirnov, 1998). In order to produce reliable data about the eJective tokamak energy emission, the detectors were calibrated using a 137 Cs source standard so that, above the threshold interval 30 –150 kV, the response of the TLDs were linear and independent of the absorbed energy. The angular position of the limiter, namely, the metallic diaphragm that deAnes the border of the plasma column inside the vessel, has been taken as a reference point to set the origin of the angular displacement ( = 0◦ ). The tokamak pre-ionisation Alament is located at an angle = 300◦ between the TLDs in rows 10 and 11. The experiment was performed during two series of 30 tokamak discharges. The Arst series with a plateau level plasma current IP =4 kA, and the second one with IP =8 kA. The duration of each discharge was approximately 600 s.
3. Results and discussion Fig. 1 shows some representative samples of the general behaviour of X-ray emission as measured from and around the tokamak vessel wall. We adopted gray (Gy) as the only
Fig. 2. Typical X-ray dose plotted as a function of the radial and angular distribution of detectors. (a) Plasma current IP = 4000 A, (b) Plasma current IP = 8000 A.
Fig. 1. Absorbed dose (Gy) as a function of the distance from the tokamak vessel wall (m) for two levels of plasma current (4000 A, 8000 A), at the following toroidal angle directions: (a) = 0◦ , (b) = 210◦ , (c) = 330◦ .
radiation unit in this work because CaSO4 : Dy TLD detectors register the (time-integrated) dose of energy absorbed by mass unit. So, the vertical axis of the plots refers to the absorbed dose in Gy while the horizontal axis to the distance from the wall in m. The presence of maxima in the proximity of the vessel for both plasma current levels can be noticed. Further away, the curves fall to a minimum just to take a gradual ascent and Anally decay. As a starting point of the model development, all data were plotted in cylinder coordinate coaxial to the tokamak, detecting the common characteristics of their behaviour. To simplify our subsequent data processing and plotting, a Cartesian coordinate system was adopted later, where x = r cos , y = r sin and the z-axis was reserved for the absorbed dose. Figs. 2(a) and 2(b) exhibit, in an r– slab geometry for IP = 4 kA and 8 kA cases, respectively, the data set chosen to establish the models, whose points have been linked together so to help identify their behaviour. Once working in the x–y Cartesian coordinate system, two main procedures were followed. For a start, the
A. Flores et al. / Radiation Measurements 33 (2001) 409–416
experimental data were Atted through a least squares algorithm into a bicubic polynomial. Then, a more sophisticated mathematical model was examined so as to approach a physical interpretation of the phenomenon in the best possible way, leading to the completion of the models proposed in this work. The choice of a bicubic polynomial was made due to the overall radial characteristics of all the analysed dose proAles, namely: the presence of a maximum, a minimum and an asymptotic decay, to be reCected in the radial dependence of x and y variables. So, the next step was to propose a Taylor-like polynomial in the following form: P(x; y) = a1 + a2 x + a3 x2 + a4 x3 + a5 y + a6 xy + a7 x2 y + a8 x3 y + a9 y2 + a10 xy2 + a11 x2 y2 + a12 x3 y2 + a13 y3 + a14 xy3 + a15 x2 y3 + a16 x3 y3 ; where x = r cos , y = r sin . The polynomial P(x; y) represents the dose due to the radiation emitted by the tokamak. Clearly, the dose density constants ai must have dimensions according to the power of the variables they multiply. From a more physical point of view, P(x; y) can be interpreted as the superimposed contribution of several harmonics or equivalent sources of radiation, each one of them with a definite spatial Cuctuation form, and a diJerent speciAc weight ai , within the experimental conAnes. Such harmonic contributions model the rather complex distribution topology of X-ray emission, reCection and absorption, interacting within and around the tokamak vessel. In order to determine the coeLcients ai , experimental data were incorporated as constraints in the form of an overdetermined linear system A · x = b; where x = x(ai ) is a vectorial array, i = 1; 2; : : : ; 16. The results presented in Table 1 were obtained by solving this system by least squares. Rounding these results, in each case of tokamak plasma current considered, the polynomial appears as: P4 (x; y) = (25:2 Gy) + (1 Gy m−1 )x + (28:1 Gy m−1 )y − (2:7 Gy m−2 )xy − (16:9 Gy m−2 )y2 + (2:2 Gy m−3 )xy2 + (2:9 Gy m−3 )y3 − (0:5 Gy m−4 )xy3 ; P8 (x; y) = (11:3 Gy) + (1:6 Gy m−1 )x + (44:2 Gy m−1 )y − (3:8 Gy m−2 )xy − (52:1 Gy m−2 )y2 + (4:4 Gy m−3 )xy2 + (17:2 Gy m−3 )y3 − (1:4 Gy m−4 )xy3 ; the subindex on the left-hand side of the equations indicates the respective value of the current in kA. It can be observed that, within the limits of the collected data volume and with the exception of two terms, an amplitude ratio of approximately two exists between both polynomials. It is signiAcant to observe that P(x; y) is far from being isotropic and that its strongest variations occur in the
411
Table 1 Adjusted values of the polynomial coeLcients for two levels of tokamak plasma current. The coeLcients are presented in a dimensionless form for the sake of clarity CoeLcient a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16
4 kA
8 kA 10−2
2:523 × 1:0167 × 10−3 −6:86 × 10−6 1:0 × 10−8 2:8114 × 10−2 −2:7138 × 10−3 1:784 × 10−5 −2:0 × 10−8 −1:6931 × 10−2 2:1729 × 10−3 −1:455 × 10−5 2:0 × 10−8 2:9353 × 10−3 −5:2039 × 10−4 3:53 × 10−6 −5:0 × 10−9
1:1257 × 10−2 1:5750 × 10−3 −4:07 × 10−6 −5:0 × 10−7 4:4188 × 10−2 −3:8459 × 10−3 1:36 × 10−5 7:0 × 10−8 −5:2105 × 10−2 4:409 × 10−3 −2:157 × 10−5 1:0 × 10−8 1:7182 × 10−2 −1:383 × 10−3 7:7 × 10−6 −8:0 × 10−9
y-direction, mainly with linear and quadratic dependencies. This fact can be attributed to the toroidal location of the tokamak plasma limiter, whose purpose is to shear the plasma edge to a speciAc minor radius. Thus, it is highly probable that the edge plasma electrons colliding against the limiter surface emit bremsstrahlung radiation. P(x; y) behaviour is shown in Figs. 3 and 4 In them, the obtained polynomials are plotted both on x–y and r– slab meshes Aner than the original (experimental) one. Such dual geometry will be maintained in all the subsequent Agures for the sake of a clearer interpretation and every data point will be linked to its respective location predicted by the model, in order to assess the accuracy of it. Searching for an alternative, more analytical, data Atting relying on a smaller parameter set, particular attention was paid to: (a) The experimental behaviour in the 0 –2 m distance domain where the available data were more equidistantly recorded. (b) The data distribution throughout the sampled distance, especially far away from the tokamak. (c) The cross-sectional topology of the Atting at constant angles. So, it was considered that the steepest changes in the dose distribution occurred within a distance of about 3 m from the machine. Beyond it, the behaviour is assumed to be practically linear with a declining trend. The periodic nature of the close range dose reading led to the choice of sinusoidal functions to be included in the modelling, which should satisfy the observed characteristic of the number of oscillations growing inversely with the size of the plasma current.
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Fig. 3. Behaviour of the bicubic polynomial model for the case of a tokamak plasma current IP = 4000 A; (a) in x–y geometry (b) in r– geometry with the angle scaled in radians.
The fundamental sinusoidal models developed, where a, b, c, d and e are Ave suitably dimensioned Atting parameters, x = r cos , y = r sin and F is the absorbed dose in Gy: F(x; y) = [eay sin(by)][c sin(d x)] + e and F(x; y) = ay sin(bx) cos(cy) + d: Figs. 5 and 6 exhibit the behaviour of the Arst of the sinusoidal models, while Figs. 7 and 8 refer to the second one. The optimised values of the function parameters applied in Figs. 5 and 6 are presented in non-dimensional form in Table 2. Physically speaking, in assuming such functional forms we are allowing the problem to be implicitly related to a two dimensional propagation of standing waves, so as to better encompass the interaction between the radiation emission and reCection. At the same time, an attenuation functional
Fig. 4. Behaviour of the bicubic polynomial model for the case of a tokamak plasma current IP = 8000 A; (a) in x–y geometry (b) in r– geometry with the angle scaled in radians.
factor may in principle account for the overall absorption eJects. In order to work out the optimisation of the models, a Levenberg–Marquardt non-linear method was employed. According to which, a second order Taylor expansion of the squared norm of the residuals (sum of the squared diJerences between each data point and its model value) is minimised in the space of the model parameters (Barrera and Olvera, 1993). To that purpose, the so-called Mor/e algorithm (Mor/e and Wright, 1993) was applied to the problem in FORTRAN 77 language. Then, after developing more than ten increasingly reAned versions, the best results were obtained with the following one: √ k4 F(x; y) = k sin x ± 3 y + k k sin(y) y k where k = IP (IP in kA), = 0:3 m Gy1=2 , = 4:525 m−1 , = 0:43 Gy m−1 , = 1:6 Gy, = 0:4 Gy1=2 , = 0:375 Gy−1 and = 0:1 Gy=kA with the upper (+) or
A. Flores et al. / Radiation Measurements 33 (2001) 409–416
Fig. 5. Experimental data Atted according to the Arst sinusoidal model in (a) x–y geometry and (b) r– geometry, with a plasma current IP = 4000 A.
lower (−) signs applying for IP = 4 kA or 8 kA cases, respectively. Figs. 9 and 10 display the data and their predictions according to this Anal model. The functional diJerences between the two factors in square brackets point to the expected anisotropy of the tokamak radiation. As in the case of the polynomial model, the greater part of the Cuctuations takes place along the y-axis. Yet, in the present case, it is clear that the non-oscillatory dependence on y is linear, while the divergence at y = 0 is prevented by the factor sin(y). In both the polynomial and sinusoidal models it is clear that a careful compromise has been reached between the precision of the adjustment and the number of parameters involved. A global estimate of the accuracy of the models is presented in Table 3 on the basis of relative and absolute residuals as well as their norm. To gain a more at glance insight into the Atting quality of the models, Figs. 3–10 exhibit all the residuals in the form of vertical segments.
413
Fig. 6. Experimental data Atted according to the Arst sinusoidal model in (a) x–y geometry and (b) r– geometry, with a plasma current IP = 8000 A.
From them, one can notice a tendency of the large residuals to accumulate at small angles and radii. This fact points to the locally jagged radiation anisotropy created by the tokamak limiter, which is not easy for the models to follow. Not surprisingly, IP = 4 kA plasma regime turns out to be less diLcult to model due to its more even distribution of radiation. On the other hand, the sinusoidal model appears to be slightly less accurate on account of its smaller parameter set. 4. Conclusions The heuristic model processing presented in this work provides a satisfactory Atting of the backscattering phenomenon, to the extent of the data availability. In both the polynomial and sinusoidal models it is clear that a careful compromise has been reached between the precision of the adjustment and the number of parameters involved.
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Fig. 7. Experimental data Atted by the second sinusoidal model in (a) x–y and (b) r– geometry, with a plasma current IP = 4000 A.
Fig. 8. Experimental data Atted by the second sinusoidal model in (a) x–y and (b) r– geometry, with a plasma current IP = 8000 A.
Table 2 Values of the Atting parameters employed in Figs. 6 –9. Notice that their dimensions (not shown) may change according to the model Model
Plasma current (kA)
a
b
c
d
e
1
4 8
−1:1137 −0:8427
0.00514 0.003617
−316:71 2968.4
0.00779 0.00628
4.02 3.384
2
4 8
−0:4131 −1:303
5.044 5.0
3.714 6.3
non-existent
Furthermore, the formulation presented here sets the ground for a model evolution leading to an increasing consistency with the involved physical theories. Comparatively speaking, the developed sinusoidal models provide a more applicable structure with respect to the variations of plasma current and with respect to the radial distance beyond the experimental domain while requiring fewer parameters. In fact, these models depend mostly
5.355 5.362
on just one eJective parameter, as all the rest are unitary dimensional coeLcients. However, subsequent models do not necessarily invalidate the previous ones. So, the polynomial variety has proved to be simpler to manipulate provided that its structure guarantees continuous diJerentiability. Polynomials allow greater Cexibility of Atting, as they lend themselves to least square Atting techniques, and the dimensionality of their terms can be balanced without resorting to unitary coeLcients.
A. Flores et al. / Radiation Measurements 33 (2001) 409–416
Fig. 9. Final model Atting the experimental data in (a) x–y and in (b) r– geometry, for the case of a tokamak plasma current IP = 4000 A.
415
Fig. 10. Final model Atting the experimental data in (a) x–y and in (b) r– geometry, for the case of a tokamak plasma current IP = 8000 A.
Table 3 Estimated global precision of the polynomial and sinusoidal models for the IP = 4 kA and 8 kA regimes. The norm of the residuals has been calculated as the square root of the sum of the squared errors Polynomial
Norm of residuals=Gy Absolute error (max) Gy Absolute error (min)=Gy Relative error (max) Relative error (min)
Sinusoidal
IP = 4 kA
IP = 8 kA
IP = 4 kA
IP = 8 kA
0.1939 0.1010 9:0132 × 10−5 0.2025 2:4035 × 10−4
0.524 0.2284 0.0051 0.4117 0.0061
0.2272 0.1251 4:8226 × 10−4 0.2507 0.0013
1.122 0.4175 0.0021 1.7507 0.0034
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