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HT2005: Rector PhysicsT09: Thermalisation1 SLOWING DOWN OF NEUTRONS Elastic scattering of neutrons. Lethargy. Average Energy Loss per Collision. Resonance Escape Probability Neutron Spectrum in a Core.

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HT2005: Rector PhysicsT09: Thermalisation2 Chain Reaction n Moderator

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HT2005: Rector PhysicsT09: Thermalisation3 Why to Slow Down (Moderate)?

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HT2005: Rector PhysicsT09: Thermalisation4 Principles of a Nuclear Reactor n/fission N1N1 N2N2 Leakage Fast fission Resonance abs. Non-fuel abs. Leakage Non-fissile abs. Fission Slowing down Energy E 2 MeV 1 eV 200 MeV/fission 2.5

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HT2005: Rector PhysicsT09: Thermalisation5 Breeding

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HT2005: Rector PhysicsT09: Thermalisation6

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HT2005: Rector PhysicsT09: Thermalisation7 Energy Dependence

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HT2005: Rector PhysicsT09: Thermalisation8 Breeding

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HT2005: Rector PhysicsT09: Thermalisation9 Space and Energy Aspects r x y z Double differential cross section

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HT2005: Rector PhysicsT09: Thermalisation10 Differential Solid Angle x y z ezez eyey exex r d3rd3r

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HT2005: Rector PhysicsT09: Thermalisation11 Hard Sphere Model r Total scattering cross section = 2r 2 n r

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HT2005: Rector PhysicsT09: Thermalisation12 r b() impact parameter cross section ( ) d n(r)n(r) () n is the number of neutrons deflected by an angle greater than Hard Sphere Scattering

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HT2005: Rector PhysicsT09: Thermalisation13 Unit sphere r = 1 n

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HT2005: Rector PhysicsT09: Thermalisation14 Detector n Differential Cross Section

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HT2005: Rector PhysicsT09: Thermalisation15 Elastic Scattering u0u0 U0U0 U u vcvc v

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HT2005: Rector PhysicsT09: Thermalisation16 Energy Loss = 0 = 180

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HT2005: Rector PhysicsT09: Thermalisation17 E v E+dE v+dv Change of Variables Energy Velocity

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HT2005: Rector PhysicsT09: Thermalisation18 E E-dE E E0E0 E0E0 p(E;E 0 ) ??

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HT2005: Rector PhysicsT09: Thermalisation19 Quantum mechanics + detailed nuclear physics analysis conclude Elastic scattering is isotropic in CM system for: neutrons with energies E < 10 MeV light nuclei with A < 13

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HT2005: Rector PhysicsT09: Thermalisation20 E E0E0 E0E0 Post Collision Energy Distribution

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HT2005: Rector PhysicsT09: Thermalisation21 Average Logarithmic Energy Loss

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HT2005: Rector PhysicsT09: Thermalisation22 Average Logarithmic Energy Loss

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HT2005: Rector PhysicsT09: Thermalisation23

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HT2005: Rector PhysicsT09: Thermalisation24 Number of collision required for thermalisation: For non-homogeneous medium: Average cosine value of the scattering angle in CM-system

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HT2005: Rector PhysicsT09: Thermalisation25 Average Cosine in Lab-System

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HT2005: Rector PhysicsT09: Thermalisation26 MaterialA 00 1H1H100.667 2D2D20.1110.333 4 He40.3600.167 6 Li60.5100.095 9 Be90.6400.074 10 B100.6690.061 12 C120.7160.056 238 U2380.9380.003 H2OH2O**0.037 D2OD2O**0.033

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HT2005: Rector PhysicsT09: Thermalisation27 ModeratorN s s / a H2OH2O0.92719.71.3662 D2OD2O0.510360.1805860 Be0.209870.153138 C0.1581150.060166 U.00842170.00400.011 N - number of collision to thermal energy s - slowing down power s / a - moderation ratio (quality factor) Slowing-Down Features of Some Moderators

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HT2005: Rector PhysicsT09: Thermalisation28 k B = 1.38110 -23 J/K = 8.61710 -5 eV/K Velocity space: v+dv v 4v 2 dv Probability that energy level E=mv 2 /2 is occupied: Neutron Velocity Distribution

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HT2005: Rector PhysicsT09: Thermalisation29 The most probable velocity: and corresponding energy: Maxwell Distribution for Neutron Density

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HT2005: Rector PhysicsT09: Thermalisation30 Dont forget : Maxwell Distribution for Neutron Flux

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HT2005: Rector PhysicsT09: Thermalisation31

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HT2005: Rector PhysicsT09: Thermalisation32 Neutron flux distribution: For thermal neutrons Average Energy of Neutrons

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HT2005: Rector PhysicsT09: Thermalisation33 Average cosine of scattering angle: CM : LAB-system: The consequence of 0 0 in the laboratory-system is that the neutron scatters preferably forward, specially for A = 1 i.e. hydrogen and practically isotropic scattering for A = 238 i.e. Uranium, because 0 0 i.e. 90 o in average. This corresponds to isotropic scattering. tr is defined as effective mean free path for non-isotropic scattering.

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HT2005: Rector PhysicsT09: Thermalisation34 s cos s cos s t r Transport Mean Free Path Information regarding the original direction is lost

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HT2005: Rector PhysicsT09: Thermalisation35 Slowing-Down of Fast Neutrons Infinite medium Homogeneous mixture of absorbing and scattering matter Continues slowing down Uniformly distributed neutron source Q(E) (E) = [n/(cm 2 s eV)] (E)dE = number of neutrons with energies in dE about E

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HT2005: Rector PhysicsT09: Thermalisation36 E t dE dt assumed slowing-down real slowing-down Continues Slowing-Down

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HT2005: Rector PhysicsT09: Thermalisation37 q(E) - number of neutrons, which per cubic-centimeter and second pass energy E. If no absorption exists in medium, so: q(E) = Q; Q - source yield (n cm -3 s -1 ) Assuming no or weak absorption (without resonances) Neutrons of zero energy are removed from the system Energy E q(E)q(E) Slowing-Down Density E0E0 Q 0

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HT2005: Rector PhysicsT09: Thermalisation38 Lethargy Variable

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HT2005: Rector PhysicsT09: Thermalisation39 Lethargy Scale 1 collision

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HT2005: Rector PhysicsT09: Thermalisation40 Energy Lethargy E ref 0 Energy Dependence Eu q(u) u+duE+dE E/ Infinite medium, no losses, constant s

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HT2005: Rector PhysicsT09: Thermalisation41 Neutron spectrum E (E) u (u) E 0.025 eV 20 15 10 5 0 10 MeV

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HT2005: Rector PhysicsT09: Thermalisation42 Probability for absorption per collision: Number of collisions per a neutron in du or dE: Probability for absorption in du or dE: Absorption in du causes a relative change in q: Resonance Absorption Energy u E Lethargy u+du E/ uln -1 E+dE

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HT2005: Rector PhysicsT09: Thermalisation43 Resonance Escape

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HT2005: Rector PhysicsT09: Thermalisation44 (u) E u t s c c 0 (u) q0q0 (u) q

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HT2005: Rector PhysicsT09: Thermalisation45 How long time does the neutron exist under slowing-down phase respectively as thermal? Slowing-down in time - t s : Number of collisions in du: Number of collisions in dt: v (1 eV) = 1.39 10 6 cm/s v (0.1 MeV ) = 4.4 10 8 cm/s Thermal life-length - t t : Life Time

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HT2005: Rector PhysicsT09: Thermalisation46 Materialt fast ( s) t thermal ( s) H2OH2O1200 D2OD2O8 1.5 10 5 Be104300 C25 1.2 10 4 Neutrons Slowing-Down Time and Thermal Life-Time

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HT2005: Rector PhysicsT09: Thermalisation47 (1) Fission neutrons - fast neutrons (10 MeV-0.1 MeV) (2) Slowing-down neutrons resonance neutrons (0.1MeV - 1 eV) (3) Thermal neutrons (1eV - 0.) Under the Neutron Life-Time 10 MeV0.1 MeV1 eV0 (1)(2)(3) E

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HT2005: Rector PhysicsT09: Thermalisation48 The END

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HT2005: Rector PhysicsT09: Thermalisation49 = 0 = 180 E v E+dE v+dv