- Email: [email protected]
Catalogue to select the initial guess spectrum during unfolding
Nuclear Instruments and Methods in Physics Research A 476 (2002) 270–272
Catalogue to select the initial guess spectrum during unfolding * He! ctor Rene! Vega Carrilloa,*, M. Pilar Iniguez de la Torreb a
! Centro Regional de Estudios Nucleares, Universidad Autonoma de Zacatecas, Apdo. Postal 336, 98068 Zacatecas, Zac. Mexico b ! ! Departamento de F!ısica Teorica y Atomica, Molecular y Nuclear, Universidad de Valladolid, Valladolid, Spain
Abstract A new method to select the initial guess spectrum is presented. Neutron spectra unfolded from Bonner sphere data are dependent on the initial guess spectrum used in the unfolding code. The method is based on a catalogue of detector count rates calculated from a set of reported neutron spectra. The spectra of three isotopic neutron sources 252Cf, 239 PuBe and 252Cf/D2O, were measured to test the method. The unfolding was carried out using the three initial guess options included in the BUNKIUT code. Neutron spectra were also calculated using MCNP code. Unfolded spectra were compared with those calculated; in all the cases our method gives the best results. r 2002 Elsevier Science B.V. All rights reserved. PACS: 29.30.Hs; 29.25.Dz; 02.10.Sp; 87.53.Wz; 87.53.Pb; 87.53.Qc Keywords: Neutron spectroscopy; Unfolding; Monte Carlo; MCNP; Bonner spheres; Isotopic neutron sources
1. Introduction In the multisphere neutron spectrometer, also known as Bonner spheres [1], each detector is characterized by a response function. As the size of the moderator is increased, the peak of the response function shifts to higher neutron energies. The relationship between the response function, the detector count rate and the neutron fluence is described by the Fredholm integral equation of the first kind [2], whose discrete version is Cj ¼
N X
Rij Fi
for j ¼ 1; 2; :::: ; m
i¼1
*Corresponding author. Tel.: +52-492-2-70-43; fax: +52492-2-70-43. E-mail address: [email protected] (H.R. Vega Carrillo).
where Cj is the readings (count rates) of the jth detector, Rij is the response of the jth detector to neutrons in the ith energy interval, and Fi is the neutron fluence in the ith energy interval. Since the number of detectors, m; is smaller than the number of energy groups used to describe the spectrum, N; no unique solution exists and approximate unfolding procedures must be applied. There are several unfolding codes and most of them, except those based on Monte Carlo methods, use iterative routines that need an initial solution to start the unfolding. Examples of unfolding codes are BUNKIUT [3,4], LOHUI [5], SAND II [6], MAXED [7] and BONDI [8]. The quality of the final solution is affected by the initial guess spectrum [3–5,7–9]. In the present investigation, a method to select the initial guess spectrum has been developed.
0168-9002/02/$ -see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 1 4 6 7 - X
H.R. Vega Carrillo, M. Pilar Iniguez * de la Torre / Nuclear Instruments and Methods in Physics Research A 476 (2002) 270–272 271
2. The catalogue From neutron spectra published by IAEA [10] the expected Bonner sphere count rates were calculated. The trend of the count rates plotted against the sphere diameter allows us to use this information to select the initial guess spectrum. Several neutron spectra and their corresponding Bonner sphere count rates were collected in a catalogue. The experimental count rates are input in the catalogue where they are normalized to the count rate obtained with the 500 or 800 sphere. For each spectrum in the catalogue a parameter is calculated using m X Csj Cexpj 2 Qs ¼ ; s ¼ 1; 2; :::; ns Csj j¼1 Here, Csj is the count rate of the jth sphere produced by the sth spectrum in the catalogue, Cexpj is the count rate of the jth sphere produced by the unknown spectrum, m is the number of spheres used in the experiment and ns is the number of spectra in the catalog. Thus, the spectrum that should be used as initial guess in the unfolding will be that with the smallest Qvalue. This procedure was developed in Mathcads [11].
were used for each source, and in the third option the spectrum sugested by the catalogue was used. All experimental situations were calculated using the MCNP 4A code [13].
4. Results In Fig. 1, the 252Cf neutron energy spectra unfolded using the three BUNKIUT code options
Fig. 1. 252Cf neutron spectra unfolded using the BUNKIUT code and three initial guess spectra: Flat, Maxwellian and Catalogue, compared with the MCNP calculation.
3. Materials and methods In order to test this method the neutron spectra of three isotopic neutron sources, 252Cf, 252Cf/D2O and 239PuBe, were measured using a Bonner sphere spectrometer. The D2O moderated 252Cf, 252 Cf/D2O, is a 30-cm-diameter stainless steel container, 0.079 cm thick, filled with heavy water with the 252Cf source located at the center. During the measurements seven spheres were used: 0, 2, 3, 5, 8, 10 and 12 in. diameter with a 4 mm 4 mm 6 LiI(Eu) scintillator. All measurements were carried out at 100 cm source-to-detector distance. The unfolding was carried out using the BUNKIUT code and the UTA4 response matrix [12]. The code has three options to initiate the unfolding: a flat spectrum, a Maxwellian spectrum and a user defined spectrum. During unfolding, these options
Fig. 2. 252Cf/D2O neutron spectra unfolded using the BUNKIUT code and three initial guess spectra: Flat, Maxwellian and Catalogue, compared with the MCNP calculation.
272 H.R. Vega Carrillo, M. Pilar Iniguez * de la Torre / Nuclear Instruments and Methods in Physics Research A 476 (2002) 270–272
the spectra obtained with this method show the best agreement with the MCNP calculations.
Acknowledgements This work was partially supported by CONACyT (Mexico) under contracts: 1835P and 31288U.
References Fig. 3. 239PuBe neutron spectra unfolded using the BUNKIUT code and three initial guess spectra: Flat, Maxwellian and Catalogue, compared with the MCNP calculation.
are shown together with the MCNP result. The same plots for 252Cf/D2O and 239PuBe are shown in Figs. 2 and 3, respectively. In all cases it is obvious that a better unfolding performance is obtained when the initial guess suggested by the catalogue is used.
5. Conclusions The application of iterative procedures to solve an ill-posed problem needs an initial guess spectrum. Here, a method to choose the initial guess spectrum has been developed and tested. The method is based upon a set of published neutron spectra for which Bonner sphere readings have been calculated and used in a catalogue. From the plots given for each neutron source, it results that
[1] R.L. Bramblett, R.I. Ewing, T.W. Bonner, Nucl. Instr. and Meth. 9 (1960) 1. [2] M. Awschalom, R.S. Sanna, Radiat. Prot. Dosim. 10 (1985) 89. [3] K.A. Lowry, T.L. Johnson, Naval Research Laboratory, NRC Memorandum Report 5340, 1984. [4] K.A. Lowry, T.L. Johnson, Health Phys. 47 (1984) 587. [5] J.T. Routti, J.V. Sandberg, Comput. Phys. Commun. 21 (1980) 119. [6] W.N. McElroy, S. Berg, T. Crockett, R. Hawkins, Air Force Weapons Laboratory, Report AFWL-TR-67-41, 1967. [7] M. Reginatto, P. Goldhagen, Health Phys. 77 (1999) 579. [8] B. Mukherjee, Nucl. Instr. and Meth. A 432 (1999) 305. [9] D.E. Garc!ıa, H.R. Vega-Carrillo, G. Miramontes, L.E. McBride, IEEE Trans. Nucl. Sci. 46 (1999) 28. [10] R.V. Griffith, J. Palfalvi, U. Madhvanth, Compendium of neutron spectra and detector responses for radiation protection purposes, IAEA Tech. Report Series No. 318, 1990. [11] MathSoft, Inc, Mathcad: User’s guide Windows version, Mathsoft, Inc., 1992. [12] N.E. Hertel, J.W. Davidson, Nucl. Instr. and Meth. A 238 (1985) 509. [13] J.F. Briesmeister, MCNP-A general Monte Carlo Nparticle transport code, Los Alamos National Laboratory Report LA-12625, 1993.
Catalogue to select the initial guess spectrum during unfolding * He! ctor Rene! Vega Carrilloa,*, M. Pilar Iniguez de la Torreb a
! Centro Regional de Estudios Nucleares, Universidad Autonoma de Zacatecas, Apdo. Postal 336, 98068 Zacatecas, Zac. Mexico b ! ! Departamento de F!ısica Teorica y Atomica, Molecular y Nuclear, Universidad de Valladolid, Valladolid, Spain
Abstract A new method to select the initial guess spectrum is presented. Neutron spectra unfolded from Bonner sphere data are dependent on the initial guess spectrum used in the unfolding code. The method is based on a catalogue of detector count rates calculated from a set of reported neutron spectra. The spectra of three isotopic neutron sources 252Cf, 239 PuBe and 252Cf/D2O, were measured to test the method. The unfolding was carried out using the three initial guess options included in the BUNKIUT code. Neutron spectra were also calculated using MCNP code. Unfolded spectra were compared with those calculated; in all the cases our method gives the best results. r 2002 Elsevier Science B.V. All rights reserved. PACS: 29.30.Hs; 29.25.Dz; 02.10.Sp; 87.53.Wz; 87.53.Pb; 87.53.Qc Keywords: Neutron spectroscopy; Unfolding; Monte Carlo; MCNP; Bonner spheres; Isotopic neutron sources
1. Introduction In the multisphere neutron spectrometer, also known as Bonner spheres [1], each detector is characterized by a response function. As the size of the moderator is increased, the peak of the response function shifts to higher neutron energies. The relationship between the response function, the detector count rate and the neutron fluence is described by the Fredholm integral equation of the first kind [2], whose discrete version is Cj ¼
N X
Rij Fi
for j ¼ 1; 2; :::: ; m
i¼1
*Corresponding author. Tel.: +52-492-2-70-43; fax: +52492-2-70-43. E-mail address: [email protected] (H.R. Vega Carrillo).
where Cj is the readings (count rates) of the jth detector, Rij is the response of the jth detector to neutrons in the ith energy interval, and Fi is the neutron fluence in the ith energy interval. Since the number of detectors, m; is smaller than the number of energy groups used to describe the spectrum, N; no unique solution exists and approximate unfolding procedures must be applied. There are several unfolding codes and most of them, except those based on Monte Carlo methods, use iterative routines that need an initial solution to start the unfolding. Examples of unfolding codes are BUNKIUT [3,4], LOHUI [5], SAND II [6], MAXED [7] and BONDI [8]. The quality of the final solution is affected by the initial guess spectrum [3–5,7–9]. In the present investigation, a method to select the initial guess spectrum has been developed.
0168-9002/02/$ -see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 1 4 6 7 - X
H.R. Vega Carrillo, M. Pilar Iniguez * de la Torre / Nuclear Instruments and Methods in Physics Research A 476 (2002) 270–272 271
2. The catalogue From neutron spectra published by IAEA [10] the expected Bonner sphere count rates were calculated. The trend of the count rates plotted against the sphere diameter allows us to use this information to select the initial guess spectrum. Several neutron spectra and their corresponding Bonner sphere count rates were collected in a catalogue. The experimental count rates are input in the catalogue where they are normalized to the count rate obtained with the 500 or 800 sphere. For each spectrum in the catalogue a parameter is calculated using m X Csj Cexpj 2 Qs ¼ ; s ¼ 1; 2; :::; ns Csj j¼1 Here, Csj is the count rate of the jth sphere produced by the sth spectrum in the catalogue, Cexpj is the count rate of the jth sphere produced by the unknown spectrum, m is the number of spheres used in the experiment and ns is the number of spectra in the catalog. Thus, the spectrum that should be used as initial guess in the unfolding will be that with the smallest Qvalue. This procedure was developed in Mathcads [11].
were used for each source, and in the third option the spectrum sugested by the catalogue was used. All experimental situations were calculated using the MCNP 4A code [13].
4. Results In Fig. 1, the 252Cf neutron energy spectra unfolded using the three BUNKIUT code options
Fig. 1. 252Cf neutron spectra unfolded using the BUNKIUT code and three initial guess spectra: Flat, Maxwellian and Catalogue, compared with the MCNP calculation.
3. Materials and methods In order to test this method the neutron spectra of three isotopic neutron sources, 252Cf, 252Cf/D2O and 239PuBe, were measured using a Bonner sphere spectrometer. The D2O moderated 252Cf, 252 Cf/D2O, is a 30-cm-diameter stainless steel container, 0.079 cm thick, filled with heavy water with the 252Cf source located at the center. During the measurements seven spheres were used: 0, 2, 3, 5, 8, 10 and 12 in. diameter with a 4 mm 4 mm 6 LiI(Eu) scintillator. All measurements were carried out at 100 cm source-to-detector distance. The unfolding was carried out using the BUNKIUT code and the UTA4 response matrix [12]. The code has three options to initiate the unfolding: a flat spectrum, a Maxwellian spectrum and a user defined spectrum. During unfolding, these options
Fig. 2. 252Cf/D2O neutron spectra unfolded using the BUNKIUT code and three initial guess spectra: Flat, Maxwellian and Catalogue, compared with the MCNP calculation.
272 H.R. Vega Carrillo, M. Pilar Iniguez * de la Torre / Nuclear Instruments and Methods in Physics Research A 476 (2002) 270–272
the spectra obtained with this method show the best agreement with the MCNP calculations.
Acknowledgements This work was partially supported by CONACyT (Mexico) under contracts: 1835P and 31288U.
References Fig. 3. 239PuBe neutron spectra unfolded using the BUNKIUT code and three initial guess spectra: Flat, Maxwellian and Catalogue, compared with the MCNP calculation.
are shown together with the MCNP result. The same plots for 252Cf/D2O and 239PuBe are shown in Figs. 2 and 3, respectively. In all cases it is obvious that a better unfolding performance is obtained when the initial guess suggested by the catalogue is used.
5. Conclusions The application of iterative procedures to solve an ill-posed problem needs an initial guess spectrum. Here, a method to choose the initial guess spectrum has been developed and tested. The method is based upon a set of published neutron spectra for which Bonner sphere readings have been calculated and used in a catalogue. From the plots given for each neutron source, it results that
[1] R.L. Bramblett, R.I. Ewing, T.W. Bonner, Nucl. Instr. and Meth. 9 (1960) 1. [2] M. Awschalom, R.S. Sanna, Radiat. Prot. Dosim. 10 (1985) 89. [3] K.A. Lowry, T.L. Johnson, Naval Research Laboratory, NRC Memorandum Report 5340, 1984. [4] K.A. Lowry, T.L. Johnson, Health Phys. 47 (1984) 587. [5] J.T. Routti, J.V. Sandberg, Comput. Phys. Commun. 21 (1980) 119. [6] W.N. McElroy, S. Berg, T. Crockett, R. Hawkins, Air Force Weapons Laboratory, Report AFWL-TR-67-41, 1967. [7] M. Reginatto, P. Goldhagen, Health Phys. 77 (1999) 579. [8] B. Mukherjee, Nucl. Instr. and Meth. A 432 (1999) 305. [9] D.E. Garc!ıa, H.R. Vega-Carrillo, G. Miramontes, L.E. McBride, IEEE Trans. Nucl. Sci. 46 (1999) 28. [10] R.V. Griffith, J. Palfalvi, U. Madhvanth, Compendium of neutron spectra and detector responses for radiation protection purposes, IAEA Tech. Report Series No. 318, 1990. [11] MathSoft, Inc, Mathcad: User’s guide Windows version, Mathsoft, Inc., 1992. [12] N.E. Hertel, J.W. Davidson, Nucl. Instr. and Meth. A 238 (1985) 509. [13] J.F. Briesmeister, MCNP-A general Monte Carlo Nparticle transport code, Los Alamos National Laboratory Report LA-12625, 1993.