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Application of the Monte Carlo method to determine the thickness of a ΔE-detector
660
Nuclear Instruments and Methods in Physics Research A281 (1989) 660-662 North-Holland, Amsterdam
Letter to the Editor APPLICATION OF THE MONTE CARLO METHOD TO DETERMINE THE THICKNESS OF A DE-DETECTOR Peter KOZMA Joint Institute for Nuclear Research, Dubna, USSR Received 22 February 1989
Calculations employing the simple Monte Carlo model of gamma-ray interaction in a silicon t1E-detector provide a very reliable energy spectrum of Compton-scattered electrons . This makes it possible to determine the thickness of a Si A E-detector with a good accuracy .
The thickness of silicon AE-detectors can be measured with a calibrated gamma-ray source using Compton conversion [1] for which the cross section is known . The spectrum produced by a 137Cs gamma-ray source (662 keV) in a Si AE-detector with a radius of 16 mm and a thickness of about 500 [ .m is displayed in fig . 1 . The dotted curve is the experimental Compton continuum whose edge appears at 478 keV, the full line represents the Klein-Nishina theoretical shape of the Compton spectrum . It is seen that the simple
0
Fz
0 U
0
50
100
150 CHANNEL Fig . 1 . Spectrum produced by a 137CS gamma-ray source in the silicon 1>E-detector. The full line is the Klein-Nishina distribution . 0168-9002/89/$03 .50 © Elsevier Science Publishers B .V . (North-Holland Physics Publishing Division)
Klein-Nishina distribution does not adequately fit the measured spectrum for an actual experimental arrangement . It should be noted that the theoretical distribution does not contain contributions from multiply scattered photons in a finite detector. However, a correction for Compton electrons escaping through the edge of the detector must be introduced . Moreover, an attentive approximation of the part of the spectrum missing due to the threshold effects of electronics is also needed . For these reasons we have tried to fit the experimental energy spectra of Compton-scattered electrons by the function calculated using the Monte Carlo method . We have used the Monte Carlo model of scattering and absorption events of a gamma-ray in a finite homogeneous medium as described in ref. [2] . The geometry of a real experimental arrangement considered in the calculation is illustrated in fig. 2 . A gamma-ray history starts by selecting a random initial direction 00 , $o from an isotropic source Ns , and the flight distance l o . Then the Cartesian coordinates of the entry point No (x o , yo, z o = 0) and the point of the first interaction are computed . The type of interaction is determined from the comparison of the ratio of the photoelectric and total absorption coefficients for initial energy Eo to a generated random number. If an event of photoelectric origin is produced, a given gamma-ray history is terminated . For a Compton-scattering event, the Klein-Nishina distribution is sampled to determine the angle of the scattered gamma-ray and the history is continued. After evaluation of the scattering angle the values of photon and electron energies, as well as the polar angle of the scattered electron and its path length, can be calculated (see below) . In general, the state of the gamma-ray after the n th scattering, Sn = (En , On , 0, xn, yn, z n ), is specified by a set of quantities : energy E, coordinates 0, q) describing the direction of motion in a spherical coordinate system with the z-axis as a
661
P. Kozma / Thickness ojd E-detector
where p(En) and Q are the total attenuation coefficient [3] for a gamma-ray of energy E and a random number, respectively . (3) Angular coordinates : B, 1 = arccos[cos 0. cos w,,, +sin 0. sin w,,, cos X, + ,], sin Xn+1 sin wn1 + (P + , _ O + arcsin sin Bn+1 1 where Xn+1 is the randomly selected azimuthal scattering angle (see fig. 3) distributed uniformly between 0 and 2 iT . The state of a Compton electron after the (n + 1)th scattering is specified by its energy En+ t and the polar angle w;"+ 1 with respect to the direction of motion of the gamma-ray. These quantities are given as a function of the energy En+1 of the gamma-ray and the scattering angle w,Y, +1 of the photon after the (n + 1)th scattering :
z Fig. 2. Geometry considered in the calculation . polar one (see fig. 3), and Cartesian coordinates x, y, z; So = (Eo, 00, q)o, xo, yo, zo = 0) being the state of a photon entering the detector. The steps in the calculation of S +1 quantities, given Sn, are: (1) Energy : En
En+1 =
1 + En/m o c 2 (1 - Cos wn+1)
where m o c 2 = 511 keV and the photon scattering angle wn+1 is sampled from the theoretical distribution . (2) Cartesian coordinates : xn+1 = x n + 1 sin 0. cos e ,
y,,,
Z n+1
yn
= zn
Ene1+ 1
_ aEn+1 ( 1 - COs wn + 1) -
(5)
1+a(1-COs wn+1)
wei n+1 = arctan f
(1
I , + a ) tan( w +1/2),
(6)
where a = E +1/moc 2 . The path length of a scattered electron is obtained from the range R e, of the electron with energy Eel [MeV] in silicon [4] : R e, = 0 .1768Eél' 265-0.0954Eri [cm] .
In the case of the path length exceeding the dimension
+ 1 . sin 0. sin + In cos 0,
with the flight distance l determined from the inverse probability distribution In
_
1 1 It(En)1-Q,
103 hz
0 U
0
Z
50 x
Fig. 3. Angular representation in the spherical coordinate frame.
100
150
CHANNEL Fig. 4. Resultant fit of the experimental spectrum by the Monte Carlo model curve.
662
P. Kozma / Thickness of A E-detector
of a cylindrical detector in a given direction, the position of electron escape is calculated, and only an appropriate part of its energy is registered. As a result of the procedure described here, the energy spectrum of Compton-scattered electrons corrected for the escape through the edge of the detector is obtained. It should be noted that the results of the calculation are only of random character . A limit on the accuracy of any set of Monte Carlo results is a statistical uncertainty which can be reduced by increasing the number of histories. Fig . 4 shows the experimental spectrum compared to the Monte Carlo model calculations for a t37 Cs gammaray source and the real experimental arrangement . As can be seen, the calculated curve reproduces the shape of the measured Compton profile in a silicon AE-detector about 500 wm in thickness better than the Klein-Nishina theoretical distribution . It is evident that the corrected calculated curve can also produce a good approximation of the missing part of the experimental spectrum below the electronic threshold . One can conclude that the calculations employing the simple Monte Carlo model of gamma-ray interaction in a silicon AE-
detector provide a very reliable spectrum of scattered electrons . The least-squares fit of this spectrum to an experimental Compton continuum, measured with a calibrated gamma-ray source, enables to determine the thickness of a Si AE-detector with a good accuracy .
Acknowledgements The author is grateful to Prof . V .A . Nikitin and Prof . I .A . Savin for their interest in this work and a critical reading of the manuscript.
References [1] R .G . Miller and R.W. Kavanagh, Nucl . Instr . and Meth. 48 (1967) 13 . [2] P . Kozma, P. B6m and J . Vincour, Czech . J . Phys . B30 (1980) 503 . [3] W .M.J . Veigele, Atomic Data Tables 5 (1973) 51 . [4] R .P . Gardner, A .M . Yacout, J. Zhang and K . Verghese, Nucl . Instr . and Meth . A242 (1986) 399 .
Nuclear Instruments and Methods in Physics Research A281 (1989) 660-662 North-Holland, Amsterdam
Letter to the Editor APPLICATION OF THE MONTE CARLO METHOD TO DETERMINE THE THICKNESS OF A DE-DETECTOR Peter KOZMA Joint Institute for Nuclear Research, Dubna, USSR Received 22 February 1989
Calculations employing the simple Monte Carlo model of gamma-ray interaction in a silicon t1E-detector provide a very reliable energy spectrum of Compton-scattered electrons . This makes it possible to determine the thickness of a Si A E-detector with a good accuracy .
The thickness of silicon AE-detectors can be measured with a calibrated gamma-ray source using Compton conversion [1] for which the cross section is known . The spectrum produced by a 137Cs gamma-ray source (662 keV) in a Si AE-detector with a radius of 16 mm and a thickness of about 500 [ .m is displayed in fig . 1 . The dotted curve is the experimental Compton continuum whose edge appears at 478 keV, the full line represents the Klein-Nishina theoretical shape of the Compton spectrum . It is seen that the simple
0
Fz
0 U
0
50
100
150 CHANNEL Fig . 1 . Spectrum produced by a 137CS gamma-ray source in the silicon 1>E-detector. The full line is the Klein-Nishina distribution . 0168-9002/89/$03 .50 © Elsevier Science Publishers B .V . (North-Holland Physics Publishing Division)
Klein-Nishina distribution does not adequately fit the measured spectrum for an actual experimental arrangement . It should be noted that the theoretical distribution does not contain contributions from multiply scattered photons in a finite detector. However, a correction for Compton electrons escaping through the edge of the detector must be introduced . Moreover, an attentive approximation of the part of the spectrum missing due to the threshold effects of electronics is also needed . For these reasons we have tried to fit the experimental energy spectra of Compton-scattered electrons by the function calculated using the Monte Carlo method . We have used the Monte Carlo model of scattering and absorption events of a gamma-ray in a finite homogeneous medium as described in ref. [2] . The geometry of a real experimental arrangement considered in the calculation is illustrated in fig. 2 . A gamma-ray history starts by selecting a random initial direction 00 , $o from an isotropic source Ns , and the flight distance l o . Then the Cartesian coordinates of the entry point No (x o , yo, z o = 0) and the point of the first interaction are computed . The type of interaction is determined from the comparison of the ratio of the photoelectric and total absorption coefficients for initial energy Eo to a generated random number. If an event of photoelectric origin is produced, a given gamma-ray history is terminated . For a Compton-scattering event, the Klein-Nishina distribution is sampled to determine the angle of the scattered gamma-ray and the history is continued. After evaluation of the scattering angle the values of photon and electron energies, as well as the polar angle of the scattered electron and its path length, can be calculated (see below) . In general, the state of the gamma-ray after the n th scattering, Sn = (En , On , 0, xn, yn, z n ), is specified by a set of quantities : energy E, coordinates 0, q) describing the direction of motion in a spherical coordinate system with the z-axis as a
661
P. Kozma / Thickness ojd E-detector
where p(En) and Q are the total attenuation coefficient [3] for a gamma-ray of energy E and a random number, respectively . (3) Angular coordinates : B, 1 = arccos[cos 0. cos w,,, +sin 0. sin w,,, cos X, + ,], sin Xn+1 sin wn1 + (P + , _ O + arcsin sin Bn+1 1 where Xn+1 is the randomly selected azimuthal scattering angle (see fig. 3) distributed uniformly between 0 and 2 iT . The state of a Compton electron after the (n + 1)th scattering is specified by its energy En+ t and the polar angle w;"+ 1 with respect to the direction of motion of the gamma-ray. These quantities are given as a function of the energy En+1 of the gamma-ray and the scattering angle w,Y, +1 of the photon after the (n + 1)th scattering :
z Fig. 2. Geometry considered in the calculation . polar one (see fig. 3), and Cartesian coordinates x, y, z; So = (Eo, 00, q)o, xo, yo, zo = 0) being the state of a photon entering the detector. The steps in the calculation of S +1 quantities, given Sn, are: (1) Energy : En
En+1 =
1 + En/m o c 2 (1 - Cos wn+1)
where m o c 2 = 511 keV and the photon scattering angle wn+1 is sampled from the theoretical distribution . (2) Cartesian coordinates : xn+1 = x n + 1 sin 0. cos e ,
y,,,
Z n+1
yn
= zn
Ene1+ 1
_ aEn+1 ( 1 - COs wn + 1) -
(5)
1+a(1-COs wn+1)
wei n+1 = arctan f
(1
I , + a ) tan( w +1/2),
(6)
where a = E +1/moc 2 . The path length of a scattered electron is obtained from the range R e, of the electron with energy Eel [MeV] in silicon [4] : R e, = 0 .1768Eél' 265-0.0954Eri [cm] .
In the case of the path length exceeding the dimension
+ 1 . sin 0. sin + In cos 0,
with the flight distance l determined from the inverse probability distribution In
_
1 1 It(En)1-Q,
103 hz
0 U
0
Z
50 x
Fig. 3. Angular representation in the spherical coordinate frame.
100
150
CHANNEL Fig. 4. Resultant fit of the experimental spectrum by the Monte Carlo model curve.
662
P. Kozma / Thickness of A E-detector
of a cylindrical detector in a given direction, the position of electron escape is calculated, and only an appropriate part of its energy is registered. As a result of the procedure described here, the energy spectrum of Compton-scattered electrons corrected for the escape through the edge of the detector is obtained. It should be noted that the results of the calculation are only of random character . A limit on the accuracy of any set of Monte Carlo results is a statistical uncertainty which can be reduced by increasing the number of histories. Fig . 4 shows the experimental spectrum compared to the Monte Carlo model calculations for a t37 Cs gammaray source and the real experimental arrangement . As can be seen, the calculated curve reproduces the shape of the measured Compton profile in a silicon AE-detector about 500 wm in thickness better than the Klein-Nishina theoretical distribution . It is evident that the corrected calculated curve can also produce a good approximation of the missing part of the experimental spectrum below the electronic threshold . One can conclude that the calculations employing the simple Monte Carlo model of gamma-ray interaction in a silicon AE-
detector provide a very reliable spectrum of scattered electrons . The least-squares fit of this spectrum to an experimental Compton continuum, measured with a calibrated gamma-ray source, enables to determine the thickness of a Si AE-detector with a good accuracy .
Acknowledgements The author is grateful to Prof . V .A . Nikitin and Prof . I .A . Savin for their interest in this work and a critical reading of the manuscript.
References [1] R .G . Miller and R.W. Kavanagh, Nucl . Instr . and Meth. 48 (1967) 13 . [2] P . Kozma, P. B6m and J . Vincour, Czech . J . Phys . B30 (1980) 503 . [3] W .M.J . Veigele, Atomic Data Tables 5 (1973) 51 . [4] R .P . Gardner, A .M . Yacout, J. Zhang and K . Verghese, Nucl . Instr . and Meth . A242 (1986) 399 .