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Compton scattering of the accelerator radiation in cylindrical cavity wall
ARTICLE IN PRESS
Optik
Optics
Optik 118 (2007) 153–157 www.elsevier.de/ijleo
Compton scattering of the accelerator radiation in cylindrical cavity wall Monika Borwin´ska, Graz˙ yna Mulak Institute of Physics, Wroc!aw University of Technology, 50–370 Wroc!aw, ul. Wybrzez˙e Wyspian´skiego 27, Poland Received 15 September 2005; accepted 8 January 2006
Abstract Starting from the Klein–Nishina relations describing Compton scattering there was calculated the intensity distribution of the radiation incoming from the cylindrical cavity wall. The calculations were carried out at first – as completely incoherent and next – assuming the possibility of the caustics arising. Its results, compared with the experimental ones, confirm the above supposition. r 2006 Elsevier GmbH. All rights reserved. Keywords: Intensity distribution; Circular caustic; Radiotherapy; Compton scattering
1. Introduction In the previously published work [1] there was expressed the opinion that complex conditions of the shaping therapeutic beam in radiotherapy (teletherapy) (i.e. extended source, heavy metallic blocks of the collimator, the additional collimators) can cause the wavefront modulations, which are responsible for the caustics arising. These last ones, in turn, can affect the irradiation distribution. A possible inflammatory reaction at the patient’s body, even if far away from the therapy area, could be associated with the caustic presence. This presumption based on the knowledge of common presence of caustics in wide frequency ranges and different kinds of waves [2] is an insufficient reason and requires to be confirmed. In order to do it, we have made simple experiment with cylindrical niche and carried out respective calculations.
Corresponding author.
E-mail address: [email protected] (M. Borwin´ska). 0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.01.014
2. Semiclassical Klein–Nishina description of Compton scattering Klein and Nishina consider Compton scattering by free electrons. Metallic cavity wall may be treated as a set of randomly acting polychromatic sources of which the spectral and directional characteristics is given in Encyklopedia fizyki [3] o , 1 þ eð1 cos WÞ _o e¼ 2, mc
o0 ¼
ð1Þ
where o is primary photon frequency, o0 is scattered photon frequency, W is scattering angle, m is electron rest mass. The Compton effect cross-sections for non-polarized and polarized primary radiation are given by Klein– Nishina relations [3] ds ¼
r20 o0 2 o o0 2 sin þ W dO, o0 o 2 o
(2)
ARTICLE IN PRESS M. Borwin´ska, G. Mulak / Optik 118 (2007) 153–157
154
where f ðWÞ ¼ ds=dO is carried out by means of (2), r is the distance from the scattering source to the observation point. The above relation results from requirement of spherical shape of scattered wave at long distance from the source.
3. Irradiation effect Ionizing radiation generated in 6 MeV Clinac has been recorded on photographic material Kodak X-OmatV (film for the therapy planning and verification). The optical density distribution was converted into intensity distribution using the Driffield–Hurter relation [5]: Fig. 1. Directional characteristics of Compton scattering obtained for some primary photon energies. Radial direction unit is equal to 2 1026 cm2 .
Fig. 2. Directional characteristics of Compton scattering obtained for various angles Y for photon energy of 0.5 MeV. Radial direction unit is equal to 2 1026 cm2 .
ds ¼
r20 o0 2 o o0 2 2 þ 4cos þ Y dO, o0 o 4 o
(2a)
where Y is the angle between the direction of primary and secondary photon polarization, r0 ¼ 2:81 1013 cm is the classical electron radius. It is worth to notice that for the non-polarized primary radiation maximal cross-section for all energy appears by W ¼ 0. With the energy decreasing, the distribution tends isotropic (Fig. 1). For the polarized primary radiation—this effect is even more easy to notice with one exception—for the polarization angle equal to p/2, when we obtain ds=dO ¼ 0 (Fig. 2). The intensity incoming in solid angle dO is given by Landau and Lifszyc [4] dI ¼ I 0
o0 jf ðWÞj2 , o r2
(3)
D ¼ D0 þ G log E,
(4)
where D is the optical density, E ¼ It is the so-called exposure, I is radiation intensity, t is exposure time and G is the slope of linear part of the photographic response. The irradiation effect recorded when the metallic block with two cylindrical holes has been put on the photographic plate and inserted in the beam area is presented in Fig. 3. The large gray rectangular is the area, which is irradiated by the beam formed by X–Y collimators. The dark rectangular with two bright circles is a trace left by the block. Surprisingly, the presence of the block, causes the growth of blackening, what is the result of photon–electron avalanches increasing. Both holes lie in the area of the uniform beam intensity (Fig. 3). The radiation incident on the photographic material, which covers the bottom of the cylinder is the sum of immediately (and perpendicularly to the bottom) incident radiation I0 and of radiation intensity I S scattered in the cylinder wall. Due to photographic response relation (3) we find IS ¼ 10ðD1 D2 Þ=G 1, I0
(5)
where D1 and D2 correspond to I 0 þ I S and I0, respectively. The relative scattered intensity (taken relative to its central value) become independent on I0. The intensity distribution at the bottom of cylinder after ‘‘subtraction’’ of immediately incident radiation has been compared with the numerical evaluation. Some remarks should be devoted to the choice of the cavity material. For light elements and for used voltages Compton effect dominates significantly. The pair creation is not too effective [3]; the photoeffect influence is removed by special envelope inside of which the photographic film is lying during irradiation. The cylinder dimensions—not too large—were chosen for two reasons. They are: the calculations economy and
ARTICLE IN PRESS M. Borwin´ska, G. Mulak / Optik 118 (2007) 153–157
155
Fig. 3. Irradiation effect recorded on photographic material under the block made of titanium (a) and (b) the optical density distribution in the cavity area.
account the changing density of active scattering sources with respect to the height caused by the electron–photon avalanches in wide beams—does not change the course of calculated curve.
5. Some geometrical considerations
Fig. 4. Relative intensity distribution for Compton scatteringincoherent case.
the photon–electron avalanches developing in wide beams and then changing the number of scattering centers with the cylinder height.
4. Experiment and incoherent summation The continuous spectrum of accelerator radiation was replaced with a set of frequencies located uniformly within 0–6 MeV range (histogram). The incoherent summation over all scattering angles, dictated by the geometry of the system, for all spectral components of Clinac radiation has been made according to (3). For comparison there has been marked also the real intensity distribution recorded on photographic material. Both results (related to their central values) are presented in Fig. 4. As is seen from comparison the discrepancy is evident. Neither the trials of introducing the correction connected with the deep seated sources nor taking into
In the case of the rotationally (axially) symmetrical system, e.g. cylinder, the congruence of straight lines (rays) leaving the points lying at the same height (directix) under the same angles creates one sheeted hyperboloid of revolution whose center coincide with the cylinder base center. This hyperboloid is the caustic surface. In special cases it degenerates into focal line coinciding with the symmetry axis. Hyperboloid, when intersects the cylinder base produces a circle caustic. The bottom of the cylinder is covered by the continuum of circular caustics. Locally the caustic is created by two near, physically undistinguished rays. Let P be the observation point lying at the base of the cylindrical cavity of radius R and height H. The disturbances incoming to point P from the photon scattered at W angle originate from points lying on the line of intersection of lateral cone surface of vertex P and 2W opening angle with the cylinder (Fig. 5). The sections AP and BP correspond to extremes of optical paths, while C 1 P ¼ C 2 P being the result of intersection of the plane perpendicular to the diameter A0 B0 with line l correspond to the inflexion points of the optical path function. In our case, we assume that the vicinities of every two inflexion points could be randomly coherent sources (mutually incoherent, spatially separated). Changing observation point position along A0 B0 we obtain the continuum of caustics covering the cylinder bottom. Setting apart the problem of coherence nature, let us calculate the examples of the caustics in a small cylinder ðR ¼ H ¼ 300lÞ caused by scattering of g photons by free electrons.
ARTICLE IN PRESS M. Borwin´ska, G. Mulak / Optik 118 (2007) 153–157
156
Fig. 5. Illustration to geometrical considerations.
6. Continuum of circular caustics Some examples of circular caustics calculations we can find, e.g. in [6,7]. In these works the evaluations are based on geometrical considerations (2-D). Two slices of caustics—first one created by ‘‘incoming’’ and second one formed by ‘‘outcoming’’ rays—should be used for solving the task. The caustic originating from 3-D sources distribution is the projection of 3-D singularity onto observation plane. Using the formulae given by Stamnes and Spjelkavik [8] we can avoid the consideration of two slices, useful in our work for particular case of incidence angle. In the works of Stamnes and Spjelkavik [8], basing on Van Kampen results [9], has been made the assumption that amplitude of wave incoming to observation point as well as wave-number are equal to 1. Avoiding this assumption we obtain contribution arising from inflexion point in stationary phase method: ! ka1 U ¼ 2pkb00 Ai (6) ð3ka3 Þ1=3 expðika0 Þ, ð3ka3 Þ1=3 where b00
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o0 ds 1 , ¼ I0 o dO r2
(6a)
an are the Taylor expansion coefficients of phase function, r the secondary source–observation point distance. In the above relation, we calculate Airy function of large negative arguments according to Abramowitz and Stegun [10]: AiðxÞ ¼ x1=4 ½f 1 ðzÞ cos z þ f 2 ðzÞ sin z,
(7)
where z ¼ 23x3=2 ,
(7a)
f 1 ðzÞ and f 2 ðzÞ are given in [10] in the form of a table. Fig. 6 shows an example of the intensity distribution obtained for 1 MeV photons for polarized and nonpolarized primary radiation.
Fig. 6. Intensity distribution for cylindrical niche obtained by (6) for W ¼ p=3, E 0 ¼ 1 MeV.
Fig. 7. Intensity distribution for cylindrical niche obtained by (6) for some primary photon energies.
The dependence of the intensity distribution vs. primary photons energy under fixed scattering angle has been presented in Fig. 7. The sequence of presented curves changes if the respective coefficients for spectrum components will be taken into account. For some scattering angles the caustics do not cover the whole bottom. We do not observe intensity in the central area (Fig. 8). We notice (Figs. 6–8) the long plateau between the huge intensity near the symmetry axis (focus) and in the vicinity of the wall (nearness of sources). These two causes are not the only reasons of the result of our calculation (see Section 7). This flat part of intensity distribution coincides qualitatively to the course of the measurement results from Fig. 4. The spectral coefficients and additional degrees of freedom which follow from the polarization state of primary radiation creates many possibilities to construct nearly arbitrary intensity distribution. For the lack of data regarding the radiation polarization for our source we illustrate this fact in Fig. 9 where polarized and nonpolarized radiation was arbitrarily taken in equal amounts. In our experiment the cavity was introduced into the field of uniform intensity. In calculation we presume
ARTICLE IN PRESS M. Borwin´ska, G. Mulak / Optik 118 (2007) 153–157
Fig. 8. Intensity distribution for cylindrical niche obtained by (6) for some Woarctg ðR=HÞ, E 0 ¼ 1 MeV, ds=dO according to (2).
157
wavelengths are of the order of atomic constant. The used model of the cavity with smooth walls is very simple. Even if we were equipped with respective mathematical instruments, tried in other frequency ranges, they will be useless. In immediate proximity of the cavity wall the quantum mechanics approach is indispensable. Two reliefs superimposed onto cylindrical surface—first deep, being the result of the mechanical treatment and the second one of the atomic scale, describe the real surface. The statistical model of such a surface should conform our qualitative results to the measured ones.
Acknowledgements The authors wish to thank the Director of Lower Silesian Oncology Center, Dr. Marek Pude"ko, for his permission to carry out the experiment and Mrs. Elz˙ bieta Pater and Dominika Oborska-Kumaszyn´ska for its realization. Special thanks are due to Dr. Stanis"aw Jab"onka, who in spite of his disease served us with his knowledge, help and advice till his last days. Fig. 9. Intensity distribution for circular niche obtained by (6) for W ¼ p=3, E 0 ¼ 1 MeV.
normal incidence. According to Jaworskij and Dietlaf [11] for low g radiation frequencies and oblique incidence the reflection should be taken into account (great albedo).
7. Concluding remarks It is worth noticing that in real intensity distribution we do not observe linear focus for r ¼ 0. The intensity near the wall in spite of the nearness of the sources is also not as great as calculated. The real intensities, even if great, should be finite. The used asymptotics behave singularly and should be replaced by the others. In the case of high degree of coherence, e.g. the radio- or microwave frequencies, description of the modes (eigenfunctions) traveling near the smooth wall is made by means of ‘‘whispering modes’’ [5,6], the linear focus description requires Pearcey function to be used. Our situation is much more complicated. The considered
References [1] M. Borwin´ska, G. Mulak, Causes of scalds in radiotherapy, Optik 113 (8) (2002) 343–347. [2] Yu.A. Kravtsov, Progress in Optics, vol. 26, NorthHolland, Amsterdam, New York, Oxford, 1993. [3] Encyklopedia fizyki, t.1-3, PWN, Warszawa, 1972. [4] L. Landau, E. Lifszyc, Mechanika kwantowa, teoria nierelatywistyczna, PWN, Warszawa, 1958. [5] M. Born, E. Wolf, Principles of Optics, Pergamon Press, Oxford, New York, Beijing, Toronto, 1991. [6] Yu.A. Kravtsov, Yu.I. Orlov, Caustics, Catastrophes and Wave Fields, Springer, Berlin, Heidelberg, 1993. [7] V.M. Babich, V.S. Buldyrev, Asimptoticeskie metody v zadacach difrakcii korotkich voln, Metod etalonnych zadac, Nauka, Moskva, 1972. [8] J.J. Stamnes, B. Spjelkavik, Evaluation of the field near a cusp of a caustic, Opt. Acta 30 (9) (1983) 1331–1358. [9] N.G. Van Kampen, Asymptotic treatment of diffraction problems, Physica 14 (1949) 575–589. [10] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Nauka, Moskva, 1979. [11] B.M. Jaworskij, A.A. Diet"af, Spravocnik po fizike, Nauka, Moscow, 1968.
Optik
Optics
Optik 118 (2007) 153–157 www.elsevier.de/ijleo
Compton scattering of the accelerator radiation in cylindrical cavity wall Monika Borwin´ska, Graz˙ yna Mulak Institute of Physics, Wroc!aw University of Technology, 50–370 Wroc!aw, ul. Wybrzez˙e Wyspian´skiego 27, Poland Received 15 September 2005; accepted 8 January 2006
Abstract Starting from the Klein–Nishina relations describing Compton scattering there was calculated the intensity distribution of the radiation incoming from the cylindrical cavity wall. The calculations were carried out at first – as completely incoherent and next – assuming the possibility of the caustics arising. Its results, compared with the experimental ones, confirm the above supposition. r 2006 Elsevier GmbH. All rights reserved. Keywords: Intensity distribution; Circular caustic; Radiotherapy; Compton scattering
1. Introduction In the previously published work [1] there was expressed the opinion that complex conditions of the shaping therapeutic beam in radiotherapy (teletherapy) (i.e. extended source, heavy metallic blocks of the collimator, the additional collimators) can cause the wavefront modulations, which are responsible for the caustics arising. These last ones, in turn, can affect the irradiation distribution. A possible inflammatory reaction at the patient’s body, even if far away from the therapy area, could be associated with the caustic presence. This presumption based on the knowledge of common presence of caustics in wide frequency ranges and different kinds of waves [2] is an insufficient reason and requires to be confirmed. In order to do it, we have made simple experiment with cylindrical niche and carried out respective calculations.
Corresponding author.
E-mail address: [email protected] (M. Borwin´ska). 0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.01.014
2. Semiclassical Klein–Nishina description of Compton scattering Klein and Nishina consider Compton scattering by free electrons. Metallic cavity wall may be treated as a set of randomly acting polychromatic sources of which the spectral and directional characteristics is given in Encyklopedia fizyki [3] o , 1 þ eð1 cos WÞ _o e¼ 2, mc
o0 ¼
ð1Þ
where o is primary photon frequency, o0 is scattered photon frequency, W is scattering angle, m is electron rest mass. The Compton effect cross-sections for non-polarized and polarized primary radiation are given by Klein– Nishina relations [3] ds ¼
r20 o0 2 o o0 2 sin þ W dO, o0 o 2 o
(2)
ARTICLE IN PRESS M. Borwin´ska, G. Mulak / Optik 118 (2007) 153–157
154
where f ðWÞ ¼ ds=dO is carried out by means of (2), r is the distance from the scattering source to the observation point. The above relation results from requirement of spherical shape of scattered wave at long distance from the source.
3. Irradiation effect Ionizing radiation generated in 6 MeV Clinac has been recorded on photographic material Kodak X-OmatV (film for the therapy planning and verification). The optical density distribution was converted into intensity distribution using the Driffield–Hurter relation [5]: Fig. 1. Directional characteristics of Compton scattering obtained for some primary photon energies. Radial direction unit is equal to 2 1026 cm2 .
Fig. 2. Directional characteristics of Compton scattering obtained for various angles Y for photon energy of 0.5 MeV. Radial direction unit is equal to 2 1026 cm2 .
ds ¼
r20 o0 2 o o0 2 2 þ 4cos þ Y dO, o0 o 4 o
(2a)
where Y is the angle between the direction of primary and secondary photon polarization, r0 ¼ 2:81 1013 cm is the classical electron radius. It is worth to notice that for the non-polarized primary radiation maximal cross-section for all energy appears by W ¼ 0. With the energy decreasing, the distribution tends isotropic (Fig. 1). For the polarized primary radiation—this effect is even more easy to notice with one exception—for the polarization angle equal to p/2, when we obtain ds=dO ¼ 0 (Fig. 2). The intensity incoming in solid angle dO is given by Landau and Lifszyc [4] dI ¼ I 0
o0 jf ðWÞj2 , o r2
(3)
D ¼ D0 þ G log E,
(4)
where D is the optical density, E ¼ It is the so-called exposure, I is radiation intensity, t is exposure time and G is the slope of linear part of the photographic response. The irradiation effect recorded when the metallic block with two cylindrical holes has been put on the photographic plate and inserted in the beam area is presented in Fig. 3. The large gray rectangular is the area, which is irradiated by the beam formed by X–Y collimators. The dark rectangular with two bright circles is a trace left by the block. Surprisingly, the presence of the block, causes the growth of blackening, what is the result of photon–electron avalanches increasing. Both holes lie in the area of the uniform beam intensity (Fig. 3). The radiation incident on the photographic material, which covers the bottom of the cylinder is the sum of immediately (and perpendicularly to the bottom) incident radiation I0 and of radiation intensity I S scattered in the cylinder wall. Due to photographic response relation (3) we find IS ¼ 10ðD1 D2 Þ=G 1, I0
(5)
where D1 and D2 correspond to I 0 þ I S and I0, respectively. The relative scattered intensity (taken relative to its central value) become independent on I0. The intensity distribution at the bottom of cylinder after ‘‘subtraction’’ of immediately incident radiation has been compared with the numerical evaluation. Some remarks should be devoted to the choice of the cavity material. For light elements and for used voltages Compton effect dominates significantly. The pair creation is not too effective [3]; the photoeffect influence is removed by special envelope inside of which the photographic film is lying during irradiation. The cylinder dimensions—not too large—were chosen for two reasons. They are: the calculations economy and
ARTICLE IN PRESS M. Borwin´ska, G. Mulak / Optik 118 (2007) 153–157
155
Fig. 3. Irradiation effect recorded on photographic material under the block made of titanium (a) and (b) the optical density distribution in the cavity area.
account the changing density of active scattering sources with respect to the height caused by the electron–photon avalanches in wide beams—does not change the course of calculated curve.
5. Some geometrical considerations
Fig. 4. Relative intensity distribution for Compton scatteringincoherent case.
the photon–electron avalanches developing in wide beams and then changing the number of scattering centers with the cylinder height.
4. Experiment and incoherent summation The continuous spectrum of accelerator radiation was replaced with a set of frequencies located uniformly within 0–6 MeV range (histogram). The incoherent summation over all scattering angles, dictated by the geometry of the system, for all spectral components of Clinac radiation has been made according to (3). For comparison there has been marked also the real intensity distribution recorded on photographic material. Both results (related to their central values) are presented in Fig. 4. As is seen from comparison the discrepancy is evident. Neither the trials of introducing the correction connected with the deep seated sources nor taking into
In the case of the rotationally (axially) symmetrical system, e.g. cylinder, the congruence of straight lines (rays) leaving the points lying at the same height (directix) under the same angles creates one sheeted hyperboloid of revolution whose center coincide with the cylinder base center. This hyperboloid is the caustic surface. In special cases it degenerates into focal line coinciding with the symmetry axis. Hyperboloid, when intersects the cylinder base produces a circle caustic. The bottom of the cylinder is covered by the continuum of circular caustics. Locally the caustic is created by two near, physically undistinguished rays. Let P be the observation point lying at the base of the cylindrical cavity of radius R and height H. The disturbances incoming to point P from the photon scattered at W angle originate from points lying on the line of intersection of lateral cone surface of vertex P and 2W opening angle with the cylinder (Fig. 5). The sections AP and BP correspond to extremes of optical paths, while C 1 P ¼ C 2 P being the result of intersection of the plane perpendicular to the diameter A0 B0 with line l correspond to the inflexion points of the optical path function. In our case, we assume that the vicinities of every two inflexion points could be randomly coherent sources (mutually incoherent, spatially separated). Changing observation point position along A0 B0 we obtain the continuum of caustics covering the cylinder bottom. Setting apart the problem of coherence nature, let us calculate the examples of the caustics in a small cylinder ðR ¼ H ¼ 300lÞ caused by scattering of g photons by free electrons.
ARTICLE IN PRESS M. Borwin´ska, G. Mulak / Optik 118 (2007) 153–157
156
Fig. 5. Illustration to geometrical considerations.
6. Continuum of circular caustics Some examples of circular caustics calculations we can find, e.g. in [6,7]. In these works the evaluations are based on geometrical considerations (2-D). Two slices of caustics—first one created by ‘‘incoming’’ and second one formed by ‘‘outcoming’’ rays—should be used for solving the task. The caustic originating from 3-D sources distribution is the projection of 3-D singularity onto observation plane. Using the formulae given by Stamnes and Spjelkavik [8] we can avoid the consideration of two slices, useful in our work for particular case of incidence angle. In the works of Stamnes and Spjelkavik [8], basing on Van Kampen results [9], has been made the assumption that amplitude of wave incoming to observation point as well as wave-number are equal to 1. Avoiding this assumption we obtain contribution arising from inflexion point in stationary phase method: ! ka1 U ¼ 2pkb00 Ai (6) ð3ka3 Þ1=3 expðika0 Þ, ð3ka3 Þ1=3 where b00
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o0 ds 1 , ¼ I0 o dO r2
(6a)
an are the Taylor expansion coefficients of phase function, r the secondary source–observation point distance. In the above relation, we calculate Airy function of large negative arguments according to Abramowitz and Stegun [10]: AiðxÞ ¼ x1=4 ½f 1 ðzÞ cos z þ f 2 ðzÞ sin z,
(7)
where z ¼ 23x3=2 ,
(7a)
f 1 ðzÞ and f 2 ðzÞ are given in [10] in the form of a table. Fig. 6 shows an example of the intensity distribution obtained for 1 MeV photons for polarized and nonpolarized primary radiation.
Fig. 6. Intensity distribution for cylindrical niche obtained by (6) for W ¼ p=3, E 0 ¼ 1 MeV.
Fig. 7. Intensity distribution for cylindrical niche obtained by (6) for some primary photon energies.
The dependence of the intensity distribution vs. primary photons energy under fixed scattering angle has been presented in Fig. 7. The sequence of presented curves changes if the respective coefficients for spectrum components will be taken into account. For some scattering angles the caustics do not cover the whole bottom. We do not observe intensity in the central area (Fig. 8). We notice (Figs. 6–8) the long plateau between the huge intensity near the symmetry axis (focus) and in the vicinity of the wall (nearness of sources). These two causes are not the only reasons of the result of our calculation (see Section 7). This flat part of intensity distribution coincides qualitatively to the course of the measurement results from Fig. 4. The spectral coefficients and additional degrees of freedom which follow from the polarization state of primary radiation creates many possibilities to construct nearly arbitrary intensity distribution. For the lack of data regarding the radiation polarization for our source we illustrate this fact in Fig. 9 where polarized and nonpolarized radiation was arbitrarily taken in equal amounts. In our experiment the cavity was introduced into the field of uniform intensity. In calculation we presume
ARTICLE IN PRESS M. Borwin´ska, G. Mulak / Optik 118 (2007) 153–157
Fig. 8. Intensity distribution for cylindrical niche obtained by (6) for some Woarctg ðR=HÞ, E 0 ¼ 1 MeV, ds=dO according to (2).
157
wavelengths are of the order of atomic constant. The used model of the cavity with smooth walls is very simple. Even if we were equipped with respective mathematical instruments, tried in other frequency ranges, they will be useless. In immediate proximity of the cavity wall the quantum mechanics approach is indispensable. Two reliefs superimposed onto cylindrical surface—first deep, being the result of the mechanical treatment and the second one of the atomic scale, describe the real surface. The statistical model of such a surface should conform our qualitative results to the measured ones.
Acknowledgements The authors wish to thank the Director of Lower Silesian Oncology Center, Dr. Marek Pude"ko, for his permission to carry out the experiment and Mrs. Elz˙ bieta Pater and Dominika Oborska-Kumaszyn´ska for its realization. Special thanks are due to Dr. Stanis"aw Jab"onka, who in spite of his disease served us with his knowledge, help and advice till his last days. Fig. 9. Intensity distribution for circular niche obtained by (6) for W ¼ p=3, E 0 ¼ 1 MeV.
normal incidence. According to Jaworskij and Dietlaf [11] for low g radiation frequencies and oblique incidence the reflection should be taken into account (great albedo).
7. Concluding remarks It is worth noticing that in real intensity distribution we do not observe linear focus for r ¼ 0. The intensity near the wall in spite of the nearness of the sources is also not as great as calculated. The real intensities, even if great, should be finite. The used asymptotics behave singularly and should be replaced by the others. In the case of high degree of coherence, e.g. the radio- or microwave frequencies, description of the modes (eigenfunctions) traveling near the smooth wall is made by means of ‘‘whispering modes’’ [5,6], the linear focus description requires Pearcey function to be used. Our situation is much more complicated. The considered
References [1] M. Borwin´ska, G. Mulak, Causes of scalds in radiotherapy, Optik 113 (8) (2002) 343–347. [2] Yu.A. Kravtsov, Progress in Optics, vol. 26, NorthHolland, Amsterdam, New York, Oxford, 1993. [3] Encyklopedia fizyki, t.1-3, PWN, Warszawa, 1972. [4] L. Landau, E. Lifszyc, Mechanika kwantowa, teoria nierelatywistyczna, PWN, Warszawa, 1958. [5] M. Born, E. Wolf, Principles of Optics, Pergamon Press, Oxford, New York, Beijing, Toronto, 1991. [6] Yu.A. Kravtsov, Yu.I. Orlov, Caustics, Catastrophes and Wave Fields, Springer, Berlin, Heidelberg, 1993. [7] V.M. Babich, V.S. Buldyrev, Asimptoticeskie metody v zadacach difrakcii korotkich voln, Metod etalonnych zadac, Nauka, Moskva, 1972. [8] J.J. Stamnes, B. Spjelkavik, Evaluation of the field near a cusp of a caustic, Opt. Acta 30 (9) (1983) 1331–1358. [9] N.G. Van Kampen, Asymptotic treatment of diffraction problems, Physica 14 (1949) 575–589. [10] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Nauka, Moskva, 1979. [11] B.M. Jaworskij, A.A. Diet"af, Spravocnik po fizike, Nauka, Moscow, 1968.