SLOWING DOWN OF NEUTRONSElastic scattering of neutrons.Lethargy. Average Energy Loss per Collision.Resonance Escape ProbabilityNeutron Spectrum in a Core.

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Chain ReactionnModeratorModerator

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Why to Slow Down (Moderate)?

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Principles of a Nuclear Reactorn n/fissionN1N2LeakageFast fissionResonance abs.Non-fuel abs.LeakageNon-fissile abs.FissionSlowing downEnergy E2 MeV1 eV200 MeV/fission 2.5

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Breeding

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Energy Dependence

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Breeding

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Space and Energy AspectsrxyzDouble differential cross section

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Differential Solid Anglexyzezeyexrd3r

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Hard Sphere ModelrTotal scattering cross section = 2r2nr

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Hard Sphere Scatteringrb()impact parametercross section ()n(r)() n is the number of neutrons deflected by an angle greater than

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Unit sphere r = 1n

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Differential Cross SectionDetectorn

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Elastic Scatteringu0U0Uuvcvq

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Energy Loss = 0 = 180

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Change of VariablesEvE+dEv+dvEnergyVelocity

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EE-dEEaE0E0p(E;E0)??

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Quantum mechanics + detailed nuclear physics analysis concludeElastic scattering is isotropic in CM system for: neutrons with energies E < 10 MeV light nuclei with A < 13

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Post Collision Energy DistributionEaE0E0

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Average Logarithmic Energy Loss

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Average Logarithmic Energy Loss

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

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Average Cosine in Lab-System

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MaterialAm01H100.6672D20.1110.3334He40.3600.1676Li60.5100.0959Be90.6400.07410B100.6690.06112C120.7160.056238U2380.9380.003H2O**0.037D2O**0.033

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Slowing-Down Features of Some Moderators N - number of collision to thermal energyxSs - slowing down powerxSs/Sa - moderation ratio (quality factor)

ModeratorNss/aH2O0.92719.71.3662D2O0.510360.1805860Be0.209870.153138C0.1581150.060166U.00842170.00400.011

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Neutron Velocity DistributionkB = 1.38110-23 J/K = 8.61710-5 eV/KVelocity space:v+dvv4v2dvProbability that energy level E=mv2/2 is occupied:

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Maxwell Distribution for Neutron DensityThe most probable velocity:and corresponding energy:

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Maxwell Distributionfor Neutron FluxDont forget :

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Average Energy of NeutronsNeutron flux distribution:For thermal neutrons

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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. Y = 90o in average. This corresponds to isotropic scattering.ltr is defined as effective mean free path for non-isotropic scattering.

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Transport Mean Free PathYYYlscosYlscosY2lsltrInformation regarding the original direction is lost

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Slowing-Down of Fast NeutronsInfinite mediumHomogeneous mixture of absorbing and scattering matterContinues slowing downUniformly distributed neutron source Q(E)(E) = [n/(cm2seV)](E)dE = number of neutrons with energies in dE about E

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Continues Slowing-Down EtdEdtassumed slowing-downreal slowing-down

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Slowing-Down Densityq(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 (ncm-3 s-1)Assuming no or weak absorption (without resonances)Neutrons of zero energy are removed from the system

EnergyEq(E)E0Q0

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Lethargy Variable

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Lethargy Scale1 collision

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EnergyLethargyEref0Energy DependenceEuq(u)u+duE+dEE/Infinite medium, no losses, constant s

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Neutron spectrum

EF(E)uF(u)E0.025 eV2015105010 MeV

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Resonance AbsorptionProbability 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:EnergyuELethargyu+duE/uln-1E+dE

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Resonance Escape

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F(u)Eust=ss+sc~scsF0(u)q0F(u)q

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Life TimeHow long time does the neutron exist under slowing-down phase respectively as thermal?

Slowing-down in time - ts:Number of collisions in du:

Number of collisions in dt:v(1 eV) = 1.39 106 cm/s v(0.1 MeV ) = 4.4 108 cm/sThermal life-length - tt :

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Neutrons Slowing-Down Time and Thermal Life-Time

Materialtfast(ms)tthermal(ms)H2O1200D2O81.5105Be104300C251.2104

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Under the Neutron Life-Time(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.)10 MeV0.1 MeV1 eV0(1)(2)(3)E

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The END

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= 0 = 180EvE+dEv+dv

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