Advertisement

A continuum theory of scintillation in inorganic scintillating crystals

  • Fabrizio DavíEmail author
Regular Article
  • 19 Downloads

Abstract

We obtain, by starting from the balance laws of a continuum endowed with a vectorial microstructure and with a suitable thermodynamics, the evolution equation for the excitation carriers in scintillating crystals. These equations, coupled with the heat and electrostatic equations, describe the non-proportional response of a scintillator to incoming ionizing radiations in terms of a reaction and diffusion-drift system. The system of partial differential equations we arrive at allows for an explicit estimate of the decay time, a result which is obtained here for the first time for scintillators. Moreover, we show how the two most popular phenomenological models in use, namely the kinetic and diffusive models, can be recovered, among many others, as a special case of our model. An example with the available data for NaI:Tl is finally given and discussed, showing how these models depend on the energy of ionizing radiations.

Graphical abstract

Keywords

Mesoscopic and Nanoscale Systems 

References

  1. 1.
    P. Lecoq, A. Annenkov, A. Gektin, M. Korzhik, C. Pedrini, Inorganic Scintillators for Detector Systems: Physical Principles and Crystal Engineering (Springer, Switzerland, 2010) Google Scholar
  2. 2.
    J.B. Birks, D.W. Fry, L. Costrell, K. Kandiah, The Theory and Practice of Scintillation Counting, International Series of Monographs on Electronics and Instrumentation (Elsevier Science, Burlington, 2013) Google Scholar
  3. 3.
    M. Ishi, M. Kobayashi, Prog. Crystal Growth Charact. 23, 245 (1991) CrossRefGoogle Scholar
  4. 4.
    A.N. Annenkov, Y.S. Kuz’minov, Mass Growth of Large PWO4 Single Crystals for Particle Detection in High-Energy Physics Experiments at CERN (Cambridge International Science Publishing, UK, 2009) Google Scholar
  5. 5.
    C. Dujardin, E. Auffray, E. Bourret, P. Dorenbos, P. Lecoq, M. Nikl, A.N. Vasil’ev, A. Yoshikawa, R. Zhu, IEEE Trans. Nuclear Sci. 65, 1977 (2018) ADSCrossRefGoogle Scholar
  6. 6.
    W. Moses, G. Bizzarri, R.T. Williams, S.A. Payne, A.N. Vasil’ev, J. Singh, Q. Li, J.Q. Grim, W.S. Chong, IEEE Trans. Nuclear Sci. 59, 2038 (2012) ADSCrossRefGoogle Scholar
  7. 7.
    J. Singh, R.T. Williams, Eds., in Excitonic and Photonic Processes in Materials, Springer Series in Materials Science (Springer, Singapore, 2015), Vol. 203 Google Scholar
  8. 8.
    A. Vasil’ev, Microtheory of Scintillation in Crystalline Materials, in Engineering of Scintillation Materials and Radiation Technologies, Springer Proceedings in Physics 200, edited by M. Korzhik, A. Gektin (Springer, Switzerland, 2017), pp. 1–32 Google Scholar
  9. 9.
    S. Agostinelli et al., Nuclear Instrum. Methods Phys. Res. A 506, 250 (2003) ADSCrossRefGoogle Scholar
  10. 10.
    F.-X. Gentit, Nuclear Instrum. Methods Phys. Res. A 486, 35 (2002) ADSCrossRefGoogle Scholar
  11. 11.
    G. Capriz, Continua with Microstructure, Springer Tracts in Natural Philosophy (Springer Verlag, Berlin, 1989) Google Scholar
  12. 12.
    A. Vasil’ev, IEEE Trans. Nuclear Sci. 55, 1054 (2008) ADSCrossRefGoogle Scholar
  13. 13.
    G. Bizzarri, W.W. Moses, J. Singh, A.N. Vasil’ev, R.T. Williams, J. Appl. Phys. 105, 044507 (2009) ADSCrossRefGoogle Scholar
  14. 14.
    R.T. Williams, J.Q. Grim, Q. Li, K.B. Ucer, W.W. Moses, Phys. Status Solidi B 248, 426 (2011) ADSCrossRefGoogle Scholar
  15. 15.
    Q. Li, J.Q. Grim, R.T. Williams, G.A. Bizarri, W.W. Moses, J. Appl. Phys. 109, 123716 (2011) ADSCrossRefGoogle Scholar
  16. 16.
    X. Lu, Q. Li, G.A. Bizarri, K. Yang, M.R. Mayhugh, P.R. Menge, R.T. Williams, Phys. Rev. B 92, 115207 (2015) ADSCrossRefGoogle Scholar
  17. 17.
    X. Lu, S. Gridin, R.T. Williams, M.R. Mayhugh, A. Gektin, A. Syntfeld-Kazuch, L. Swiderski, M. Moszynski, Phys. Rev. Appl. 7, 014007 (2017) ADSCrossRefGoogle Scholar
  18. 18.
    Q. Li, J.Q. Grim, K.B. Ucer, A. Burger, G.A. Bizarri, W.W. Moses, R.T. Williams, Phys. Status Solidi RRL 6, 346 (2012) CrossRefGoogle Scholar
  19. 19.
    I.V. Khodyuk, P. Dorenbos, IEEE Trans. Nuclear Sci. 59, 3320 (2012) ADSCrossRefGoogle Scholar
  20. 20.
    A. Vasil’ev, A.V. Getkin, IEEE Trans. Nuclear Sci. 61, 235 (2014) ADSCrossRefGoogle Scholar
  21. 21.
    F. Daví, Light yield, decay time and reaction diffusion-drift equation in scintillators, in Proceedings of INDAM Meeting Harnack’s Inequalities and Nonlinear Operators (Springer, 2018), to appear Google Scholar
  22. 22.
    F. Daví, Thermoelastic Scintillators (2018), forthcoming Google Scholar
  23. 23.
    G. Albinus, H. Gajewski, R. Hünlich, Nonlinearity 15, 367 (2002) ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    A. Mielke, Nonlinearity 24, 1329 (2011) ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    J.E. Jaffe, D.V. Jordan, A.J. Peurrung, Nuclear Instrum.Methods Phys. Res. A580, 1378 (2007) ADSCrossRefGoogle Scholar
  26. 26.
    W.W. Ulmer, E.E. Matsinos, Eur. Phys. J. Special Topics 190, 1 (2010) ADSCrossRefGoogle Scholar
  27. 27.
    J.E. Jaffe, Nucl. Instrum. Methods Phys. Res. A570, 72 (2007) ADSCrossRefGoogle Scholar
  28. 28.
    A. Lempicki, A.J. Wojtowicz, E. Berman, Nucl. Instrum.Methods Phys. Res. A333, 304 (1993) ADSCrossRefGoogle Scholar
  29. 29.
    P.A. Rodnyi, Physical Processes in Inorganic Scintillators (CRC Press, New York, 1997) Google Scholar
  30. 30.
    H. Bethe, J. Ashkin, in Passage of Radiation Through Matter, Experimental Nuclear Physics, edited by E. Segre (John Wiley & Sons, Ltd., New York, 1952), Vol. I, pp. 166–357 Google Scholar
  31. 31.
    J.F. Ziegler, J. Appl. Phys./Rev. Appl. Phys. 85, 1249 (1999) ADSCrossRefGoogle Scholar
  32. 32.
    M. Inokuti, Rev. Mod. Phys. 43, 297 (1971) ADSCrossRefGoogle Scholar
  33. 33.
    C. Leroy, P.G. Rancoita, Principle of Radiation Interaction in Matter and Detection, 2nd edn. (World Scientific, Singapore, 2009) Google Scholar
  34. 34.
    G. Bizzarri, W.W. Moses, J. Singh, A.N. Vasil’ev, R.T. Williams, J. Lumin. 129, 1790 (2009) CrossRefGoogle Scholar
  35. 35.
    G. Bizzarri, N.J. Cherepy, W.S. Chong, G. Hull, W. Moses, S.A. Payne, J. Singh, J.D. Valentine, A.N. Vasil’ev, R.T. Williams, IEEE Trans. Nuclear Sci. 56, 2313 (2009) ADSCrossRefGoogle Scholar
  36. 36.
    W. Ulmer, Radiat. Phys. Chem. 76, 1089 (2007) ADSCrossRefGoogle Scholar
  37. 37.
    W. Ulmer, B. Schaffner, Radiat. Phys. Chem. 80, 378 (2011) ADSCrossRefGoogle Scholar
  38. 38.
    J. Singh, J. Appl. Phys. 110, 024503 (2011) ADSCrossRefGoogle Scholar
  39. 39.
    S. Gridin, A. Belsky, C. Dujardin, A. Getkin, N. Shiran, A. Vasil’ev, J. Phys. Chem. C 119, 20578 (2015) CrossRefGoogle Scholar
  40. 40.
    G. Capriz, Continua with substructure, Mesophys. Mech. (London, 2000) Google Scholar
  41. 41.
    G. Capriz, E.G. Virga, Arch. Ration. Mech. Anal. 109, 323 (1990) CrossRefGoogle Scholar
  42. 42.
    P. Podio-Guidugli, Ricerche di Matematica 55, 105 (2006) MathSciNetCrossRefGoogle Scholar
  43. 43.
    M.E. Gurtin, Physica D 92, 178 (1986) ADSCrossRefGoogle Scholar
  44. 44.
    Y. Xiao, K. Bhattacharya, Arch. Ration. Mech. Anal. 189, 59 (2008) MathSciNetCrossRefGoogle Scholar
  45. 45.
    L. Ambrosio, N. Gigli, G. Savaré, Gradient Flows in Metric Spaces and in the Spaces of Probability Measures (Birkhauser Verlag, Basel, 2005) Google Scholar
  46. 46.
    A. Mielke, On thermodynamical coupling of quantum mechanics and microscopic systems, in Proceedings of the QMath12 Conference, 2015, pp. 331–347 Google Scholar
  47. 47.
    X. Chen, A. Jüngel, Global renormalized solutions to reaction-cross diffusion systems, https://doi.org/arXiv:1771.01463v1 (2017)
  48. 48.
    J. Fischer, Nonlinear Anal. 159, 181 (2017) MathSciNetCrossRefGoogle Scholar
  49. 49.
    K. Fellner, M. Kniely, Appl. Math. Lett. 79, 196 (2018) MathSciNetCrossRefGoogle Scholar
  50. 50.
    T.R. Waite, Phys. Rev. 107, 463 (1957) ADSCrossRefGoogle Scholar
  51. 51.
    T.R. Waite, J. Chem. Phys. 28, 103 (1958) ADSCrossRefGoogle Scholar
  52. 52.
    V. Kuzovkov, E. Kotomin, Rep. Prog. Phys. 51, 1479 (1988) ADSCrossRefGoogle Scholar
  53. 53.
    E. Kotomin, V. Kuzovkov, Modern aspects of diffusion-controlled reactions. Cooperative phenomena in bimolecular processes, in Comprehensive Chemical Kinetics (Elsevier, Amsterdam, 1996), Vol. 34 Google Scholar
  54. 54.
    I.V. Khodyuk, F.G.A. Quarati, M.S. Alekhin, P. Dorenbos, J. Appl. Phys. 114, 123510 (2013) ADSCrossRefGoogle Scholar
  55. 55.
    J.Q. Grim, Q. Li, K.B. Ucer, A. Burger, G.A. Bizarri, W.W. Moses, R.T. Williams, Phys. Status Solidi A 209, 2421 (2012) ADSCrossRefGoogle Scholar
  56. 56.
    R.D. Popp, R.B. Murray, J. Phys. Chem. Solids 33, 601 (1972) ADSCrossRefGoogle Scholar
  57. 57.
    H.B. Dietrich, R.B. Murray, J. Luminescence 5, 155 (1972) ADSCrossRefGoogle Scholar
  58. 58.
    R.T. Williams, J.Q. Grim, Q. Li, K.B. Ucer, A. Burger, G.A. Bizarri, S. Kerisit, F. Gao, P. Bhattacharya, E. Tupitsyn, E. Rowe, V.M. Buliga, B. Burger, Experimental and computational results on exciton/free-carrier ratio, hot/thermalized carrier diffusion, and linear/nonlinear rate constants affecting scintillator proportionality, in Proc. SPIE 8852, Hard X-Ray, Gamma-Ray, and Neutron Detector Physics XV, 88520J, 26 September, 2013 Google Scholar
  59. 59.
    B. Henderson, G.F. Imbusch, Optical Spectroscopy of Inorganic Solids (Clarendon, Oxford, 1989) Google Scholar
  60. 60.
    P. Sibczyński, M. Moszyński, T. Szcześniak, W. Czarnacki, J. Instrum. 7, 1 (2012) Google Scholar

Copyright information

© EDP Sciences, SIF and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.DICEA, Universitá Politecnica delle MarcheAnconaItaly

Personalised recommendations