PL0000843 The Henryk Niewodniczariski Institute of Nuclear Physics Krakow, Poland REPORT No 1838/PN THERMAL NEUTRON DIFFUSION PARAMETERS DEPENDENT ON THE FLUX ENERGY DISTRIBUTION IN FINITE HYDROGENOUS MEDIA Krzysztof Drozdowicz Address: Main site: High Energy Departments, ul. Radzikowskiego 152, ul. Kawiory 26 A,, 31-342 Krakow, Poland 30-055 Krakow, Poland e-mail: [email protected] 3 1-11 e-mail: [email protected] The Henryk Niewodniczahski Institute of Nuclear Physics Krakow, Poland REPORT No 1838/PN THERMAL NEUTRON DIFFUSION PARAMETERS DEPENDENT ON THE FLUX ENERGY DISTRIBUTION IN FINITE HYDROGENOUS MEDIA Krzysztof Drozdowicz Address: ul. Radzikowskiego 152, PL- 31-342 Krakow, Poland e-mail: [email protected] Krakow 1999 PRINTED AT THE HENRYK NIEWODNICZANSKI INSTITUTE OF NUCLEAR PHYSICS KRAK6W, UL. RADZIKOWSKIEGO 152 Xerocopy: INP Krakow 27 copies. In memory of Professor Jan A. Czubek Report INP No. 1838/PN THERMAL NEUTRON DIFFUSION PARAMETERS DEPENDENT ON THE FLUX ENERGY DISTRIBUTION IN FINITE HYDROGENOUS MEDIA Krzysztof Drozdowicz The Henryk Niewodniczanski Institute of Nuclear Physics ul. Radzikowskiego 152, PL- 31-342 Krak6w, Poland ABSTRACT Macroscopic parameters for a description of the thermal neutron transport in finite volumes are considered. A very good correspondence between the theoretical and experimental parameters of hydrogenous media is attained. Thermal neutrons in the medium possess an energy distribution, which is dependent on the size (characterized by the geometric buckling) and on the neutron transport properties of the medium. In a hydrogenous material the thermal neutron transport is dominated by the scattering cross section which is strongly dependent on energy. A monoenergetic treatment of the thermal neutron group (admissible for other materials) leads in this case to a discrepancy between theoretical and experimental results. In the present paper the theoretical definitions of the pulsed thermal neutron parameters (the absorption rate, the diffusion coefficient, and the diffusion cooling coefficient) are based on Nelkin's analysis of the decay of a neutron pulse. Problems of the experimental determination of these parameters for a hydrogenous medium are discussed. A theoretical calculation of the pulsed parameters requires knowledge of the scattering kernel. For thermal neutrons it is individual for each hydrogenous material because neutron scattering on hydrogen nuclei bound in a molecule is affected by the molecular dynamics (characterized with internal energy modes which are comparable to the incident neutron energy). Granada's synthetic model for slow-neutron scattering is used. The complete up-dated formalism of calculation of the energy transfer scattering kernel after this model is presented in the paper. An influence of some minor variants within the model on the calculated differential and integral neutron parameters is shown. The theoretical energy-dependent scattering cross section (of Plexiglas) is compared to experimental results. A particular attention is paid to the calculation of the diffusion cooling coefficient. A solution of an equation, which determines the cooling function, is presented. On this basis the diffusion cooling coefficient is calculated for two hydrogenous materials (Plexiglas, polyethylene). A method is presented to calculate the thermal neutron energy-averaged (stationary and pulsed) parameters of a hydrogenous medium for which the energy dependence of the scattering cross section is unknown. New effective scattering cross sections of hydrogen are determined for this purpose. ACKNOWLEDGEMENTS This work was inspired by Prof. Jan A. Czubek (1935 - 1995). I am indebted to many people for their help or assistance at my activities that have been the basis for this paper. First, I want to mention the three institutions, in which I had the opportunity to carry on my work: my mother institute — The Henryk Niewodniczanski Institute of Nuclear Physics [Instytut Fizyki Ja.drowej im. Henryka Niewodniczanskiego] in Krakow, Poland; Department of Reactor Physics at Chalmers University of Technology [Institutionen for reaktorfysik, Chalmers tekniska hogskola] in Goteborg, Sweden; and the Neutron Division at the Bariloche Atomic Centre [Centro Atomico Bariloche] in S. C. Bariloche, Argentina. Many thanks to all my colleagues from the Neutron Transport Physics Laboratory (Dept. of Environmental and Radiation Transport Physics at the INP) for a friendly atmosphere of work, and especially to B.Sc. Jacek Burda, M.Sc. Barbara Gabanska, M.Sc. Andrzej Igielski, Mr. Tadeusz Kowalik, and Dr. Ewa Krynicka, for their participation in pulsed neutron experiments. I wish to express my particular gratitude to Assoc. Prof. Urszula Woznicka, Head of the Lab, for the fruitful collaboration and for many helpful discussions on preparation of this paper. I thank Prof. Imre Pazsit from the Chalmers University for the possibility I had to stay as the guest researcher at the Dept. of Reactor Physics. I am very indebted to Prof. Nils Goran Sjostrand from that Department for his help to organize my experiment in Studsvik (Sweden) and for the contact to physicists in the Bariloche Atomic Centre. I am grateful to Prof. J. Rolando Granada and Assoc. Prof. Victor H. Gillette from the Bariloche Atomic Centre for the close co-operation (including mutual scientific visits) and numerous discussions on details of the synthetic scattering model for slow neutrons. I thank Prof. Jerzy Loskiewicz, Head of the Department, for laborious overviewing of my typescript and advices how to make it more clear for the reader. VI CONTENTS Abstract V Acknowledgements VI Contents VII List of main symbols IX Introduction XIII 1. Basic relations for the thermal neutron transport 1 2. Thermal neutron parameters for a medium by experiment and by theory 5 2.1. The pulsed neutron experiment 5 2.2. Outline of the analysis of the decaying neutron pulse 8 2.3. Other definitions of the diffusion and diffusion cooling coefficients 13 3. Experiments on hydrogenous moderators: example of Plexiglas 16 3.1. The pulsed neutron experiment for Plexiglas 16 3.2. Measurement of the total cross section with the transmission method using monoenergetic neutrons 23 3.3. The total cross section for Plexiglas by the transmission experiment with the time-of-flight method 28 4. Approximations in the calculation of the thermal neutron parameters 29 4.1. Common approximations 31 4.1.1. Absorption 31 4.1.2. Scattering 32 4.1.3 The average cosine of the scattering angle and the transport cross section . 34 4.2. New effective scattering cross sections of hydrogen for a calculation of the energy-averaged thermal neutron parameters of hydrogenous mixtures .... 36 5. Solution of the equation for the function (J>2(£^ 42 6. Scattering kernel 48 6.1. Introduction 48 6.2. Principles of Granada's synthetic scattering model 50 6.3. The energy transfer kernel by the synthetic model 53 6.3.1. Basic characteristics of the model 54 6.3.2. The n-order terms of the scattering kernel 57 6.3.3. Calculation of the zeroth- and first-order energy transfer kernels 60 — Principal term of the zeroth-order scattering kernel 60 — Principal term of the first-order scattering kernel 61 — Phononic terms in the zeroth-order scattering kernel 62 — Phononic terms in the first-order scattering kernel 63 — The lowest frequency modes of the zeroth-order scattering kernel 64 — The lowest frequency modes of the first-order scattering kernel 65 VII 6.4. Total scattering cross section 66 — The lowest frequency modes of the total scattering cross section 67 6.5. Average cosine of the scattering angle 68 — Transport cross section 69 — Higher-order average cosines 70 7. Calculations using Granada's synthetic scattering model 71 7.1. Sensitivity of calculated thermal neutron parameters to variants in Granada's model 71 7.1.1. Variants of the energy transfer scattering kernel in the frame of the model 71 7.1.2. Calculated thermal neutron parameters on example of Plexiglas 73 — The scattering cross section and the diffusion constant 74 - The diffusion cooling coefficient 78 7.1.3. Conclusions 81 7.2. Diffusion cooling coefficient for thermal neutrons in Plexiglas 82 7.3. Diffusion cooling coefficient for thermal neutrons in polyethylene 86 8. Conclusions 90 Appendix: Free gas model as a limit case of Granada's synthetic model XIX References XXI VIII LIST OF MAIN SYMBOLS The brackets [ ] contain the number of the chapter or section in which the symbol is explained. The abbreviation GSM = Granada's synthetic scattering model. (x) = (x(E)) — general symbol for the Maxwellian-weighted average, Eq. (2.23) A — relative-to-neutron atomic mass ^moi ~ relative molecular mass fl2 - geometric buckling [Ch.l], [Sec.3.1] fl2 _ dynamic buckling [Ch.l] B2 — material buckling [Ch.l] m b(E) — average cosine of the scattering angle [Sec.2.2], [Ch.6] b — approximate value of the average cosine of the scattering angle in a hydrogenous medium C — thermal neutron diffusion cooling coefficient [Ch.2]; different theoretical definitions of the diffusion cooling coefficient: CB - Eq. (2.35) CD - Eq. (2.30) CK - Eq. (2.39) C, — from the theoretical calculation using the fit of Eq.(2.15) [Sec.2.3] C — transport correction to the diffusion cooling coefficient, Eq. (2.32) C, C, C, C — values of the diffusion cooling coefficient C according to variants of the GSM [Ch.7] C^, C — coefficients in the phononic terms of the scattering kernel and of the scattering cross section (GSM) [Ch.6] D, D(E) — thermal neutron