Chemical Engineering 412

Transcription

1 Chemical Engineering 412 Introductory Nuclear Engineering Lecture 12 Radiation/Matter Interactions II 1

2 Neutron Flux The collisions of neutrons of all energies is given by FF = ΣΣ ii 0 EE φφ EE dddd All volumetric reactions (fissions, scattering, absorption, etc.) are proportional to φφ RR ii rr = ΣΣ ii φφ rr For reactions other than with neutrons, the same equation applies, with μμ ii replacing Σ ii

3 Showing All Dependencies Flux generally depends on energy, time, and position φφ rr, EE, tt = NN pp rr, EE, tt vv EE Volumetric reaction has same dependencies RR rr, EE, tt = μμ ii rr, EE, tt φφ rr, EE, tt = ΣΣ ii rr, EE, tt φφ rr, EE, tt Commonly, neutrons are divided into energy groups by energy, so that within each group: RR rr, EE, tt dddd = ΣΣ ii rr, EE, tt φφ rr, EE, tt dddd

4 Integral values Fissions over a defined time interval F Δtt = tt 1 tt 2 VV 0 ΣΣ ii rr, EE, tt φφ rr, EE, tt dddddddddddd For time-independent properties (common apprx.) F Δtt = VV 0 ΣΣ ii rr, EE Φ rr, EE dddddddd Where the fluence, Φ rr, EE, tt tt 2φφ Φ rr, EE = rr, EE, tt dddd Δttφφ rr, EE, tt dddd tt 1

5 Fluence For time-independent properties (common apprx.) F Δtt = VV 0 ΣΣ ii rr, EE Φ rr, EE dddddddd Where the fluence, Φ rr, EE, tt tt 2φφ Φ rr, EE = rr, EE, tt dddd Δttφφ rr, EE, tt tt 1 Frequently, tt 1 is assumed to be the beginning of the reactor/fuel rod ( ) and tt 2 is current time (tt) Φ rr, EE, tt = tt φφ rr, EE, tt dddd

6 Uncollided Flux In a vacuum φφ 0 rr = SS pp 4ππrr 2 S p r P RR 0 rr = ΔVV dd RR ii rr = ΔVV dd μμ dd φφ 0 rr = ΔVV ddμμ dd SS pp 4ππrr 2

7 Uncollided Flux Through homogeneous medium φφ 0 rr = SS pp exp μμμμ 4ππrr2 S p P r RR 0 rr = ΔVV ddμμ dd SS pp 4ππrr 2 exp μμμμ

8 Uncollided Flux Through homogeneous slab t φφ 0 rr = SS pp exp μμμμ 4ππrr2 S p r P RR 0 rr = ΔVV ddμμ dd SS pp 4ππrr 2 = ΔVV ddμμ dd SS pp 4ππrr 2 exp exp μμμμ ii μμ ii tt ii

9 Fissile Nuclide Thermal Data αα = σσ γγ σσ ff νν ηη = νν σσ ff σσ aa = νν σσ ff σσ γγ +σσ ff = νν 1 + αα capture-to-fission ratio neutrons per fission neutrons per absorption

10 Quantitative Treatment Gamma cross sections σ γ = σ + σ + σ = σ + σ + pe pp C pe pp Z σ e C Attenuation coefficients µ = N σ γ = µ + µ + µ pe pp C Mass attenuation coefficients µ = ρ µ ρ µ ρ µ ρ * pe pp C µ = + +

11 Photon (γ-ray) Interactions Photoelectric effect Generally decreasing absorption with increasing energy Indicative of elemental/nuclear structure. γ-ray absorbed Low energy products Pair Production Increasing absorption with increasing energy Depends on Z (happens in Coulomb field near nucleus) γ-ray absorbed Low energy products Compton Effect Generally decreasing absorption with increasing energy Depends on electron structure and hence Z γ-ray remitted Small energy change

12 Photoelectric effect γ-ray is absorbed Photoelectric effect depends on γ-ray energy and Z electron ejection, generally from the K, L, or M levels X-ray or Auger electron emission follows electron and x-ray/auger emission generally low energy compared to γ-ray cross section designated as σ pe with interactions/volume given by INσ pe as usual Cross section depends on Z n where n is about 4, as on following graph (2 nd from here)

13 Photon Interactions in Lead

14 Photoelectric Effect

15 Pair Production Creates negatron (electron) and positron from the photon/ γ-ray Minimum energy threshold of 1.02 MeV EE mmmmmm = 2mm ee cc 2 Product particles lose kinetic energy (thermally) prior to recombining in annihilation radiation Cross section approximately proportional to Z 2

16 Pair Production

17 Compton Effect Elastic scattering of photon by an electron Scattered photon has nearly same energy as initial photon (same except for rebound energy of electron) In terms of energy EEe E' = E 1 cosϑ + E In wavelength λ λ' λ = h m c λ Serious shielding problem C = e ( ) e C ( 1 cosϑ) = 2.426x10 10 cm

18 Compton Effect

19 Total Mass Interaction Coefficients

20 Iron Mass Attenuation C

21 Electrons In Water 50 kev 1 MeV

22 In Water α particles

23 Particle Range & Stopping Power dddd dddd cccccc = ρρ ZZ AA zz22 ff II, ββ RR = RR aa ρρ aa ρρ MM MM aa PP ss = RR aa RR = ρρ MM

24 p + /e - Stopping Power

25 p + /e - Stopping Power

26 Electron Radiative Stopping Power A charged particle moving through a collection of atoms emits photons as it is deflected in the atomic fields. This is called Bremsstrahlung. The computation of this effect is complex. dddd = ρρnn aa dddd rrrrrr AA EE + mm eecc 2 ZZ 2 FF(EE, ZZ) Relativistic, heavy charged particles with rest mas MM dddd dddd rrrrrr dddd dddd cccccccc EEEE 700 mm ee MM 2

27 Fission Fragment Penetration Fission fragments travel locally

28 Fission Fusion Fission

29 Fissile vs Fissionable Critical energy = E crit = energy to cause fission If binding energy of n X(n,) n+1 X for near-zero kinetic energy electron is greater than E crit, material is fissile. Examples include 233 U, 235 U, 239 Pu, and 241 Pu. Nuclei requiring additional (typically neutron kinetic) energy are fissionable.