CH.VIII: RESONANCE REACTION RATES

1. CH.VIII: RESONANCE REACTION RATES RESONANCE CROSS SECTIONS • EFFECTIVE CROSS SECTIONS • DOPPLER EFFECT • COMPARISON WITH THE NATURAL PROFILE RESONANCE INTEGRAL RESONANCE INTEGRAL – HOMOGENEOUS THERMAL REACTORS • INFINITE DILUTION • NR AND NRIA APPROXIMATIONS RESONANCE INTEGRAL – HETEROGENEOUS THERMAL REACTORS • GEOMETRIC SELF-PROTECTION • NR AND NRIA APPROXIMATIONS • DOPPLER EFFECT

2. VIII.1 RESONANCE CROSS SECTIONS EFFECTIVE CROSS SECTIONS Cross sections (see Chap.I) given as a function of the relative velocity of the n w.r.t. the target nucleus • Impact of the thermal motion of the heavy nuclei! • No longer considered as immobile (as in chap.II)  Reaction rate: where : absolute velocities of the n and nucleus, resp. But P and : f (scalar v)  Effectivecross section: with

3. Particular cases Let 1. (see chap.I, Breit-Wigner profile for E << Eo) • Profile in the relative v unchanged in the absolute v 2. slowly variable and velocity above the thermal domain • Conservation of the relative profiles outside the resonances 3. Energy of the n low compared to the thermal zone  Effect of the thermal motion on measurements of  at low E  indep. of  !

4. DOPPLER EFFECT Rem: = convolution of and  widening of the resonance peak Doppler profile for a resonance centered in Eo >> kT ? Maxwellian spectrum for the thermal motion: Effective cross section: (Eo: energy of the relative motion !)

5. : reduced mass of the n-nucleus system Approximation: Let: : Doppler width of the peak

6. ( : peak width) and COMPARISON WITH THE NATURAL PROFILE Let:

7. and Properties of the Bethe – Placzek functions •  (low to) • Natural profiles • 0 (high to) •   Widening of the peak, but conservation of the total surface below the resonance peak (in this approximation)

8. VIII.2 RESONANCE INTEGRAL Absorption rate in a resonance peak: By definition, resonanceintegral: Flux depression in the resonance but slowing-down density +/- cst on the u of 1 unique resonance • If absorption weak or = : Before the resonance: because (as: asymptotic flux, i.e. without resonance)  I : equivalent cross section (p : scattering of potential) and

9. Resonance escape proba For a set of isolated resonances: Homogeneous mix Ex: moderator m and absorbing heavy nuclei a Heterogeneous mix Ex: fuel cell Hyp: asymptotic flux spatially constant too At first no ( scattering are different), but as = result of a large nb of collisions ( in the fuel as well as in m) • Homogenization of the cell: • Resonance escape proba V1 Vo V

10. VIII.3 RESONANCE INTEGRAL – HOMOGENEOUS THERMAL REACTORS INFINITE DILUTION Very few absorbing atoms  (u) = as(u) (resonance integral at  dilution) NR AND NRIA APPROXIMATIONS Mix of a moderator m (non-absorbing, scattering of potential m) and of N (/vol.) absorbing heavy nuclei a.

11. Microscopic cross sections per absorbing atom NR approximation (narrow resonance) Narrow resonance s.t. and i.e., in terms of moderation, qi >> ures: By definition: 

12. NRIA approximation (narrow resonance, infinite mass absorber) Narrow resonance s.t. but (resonance large enough to undergo several collisions with the absorbant  wide resonance, WR) • for the absorbant Thus Natural profile  with and

13. Remarks • NR  NRIA for    ( dilution: I  I) • I if  ([absorbant] ) • Resonance self-protection: depression of the flux reduces the value of I Doppler profile  with Remarks J(,) if (i.e. T) • Fast stabilizing effect linked to the fuel T as  E    p   keff  T  I   T

15. Choice of the approximation? Practical resonance width: p s.t. B-W>pa To compare with the mean moderation due to the absorbant p < (1 - a) Eo NR p > (1 - a) Eo NRIA Intermediate cases ? We can write with =0 (NRIA),1 (NR) • Goldstein – Cohenmethod : Intermediate value de  from the slowing-down equation

16. VIII.4 RESONANCE INTEGRAL – HETEROGENEOUS THERMAL REACTORS GEOMETRIC SELF-PROTECTION Outside resonances (see above): Asymptotic flux spatially uniform with (Rem: fuel partially moderating) In the resonances: Strong depression of the flux in Vo • Geometric self-protection of the resonance • Justification of the use of heterogeneous reactors (see notes) V1 Vo V  I 

17. NR AND NRIA APPROXIMATIONS Hyp: k(u)spatially cst in zone k; resonance  o  1 Let Pk : proba that 1 n appearing uniformly and isotropically at lethargy u in zone k will be absorbed or moderated in the other zone Slowing-down in the fuel ? NR approximation qo, q1 >>  Rem: Pk = leakage proba without collision

18. p1V1P1= to(u)VoPo Reminder chap.II Relation between Po and P1

19. Thus Wigner approximation for Po : with l : average chord length in the fuel (see appendix) NRIA approximation  since the absorbant does not moderate with

20. Thus Pk : leakage proba, with or without collision  If Pc = capture proba for 1 n emitted …, then Wigner approx: • INR, INRIA formally similar to the homogeneous case • Equivalence theorems and

21. DOPPLER EFFECT IN HETEROGENEOUS MEDIA NRcase: without Wigner, with Doppler Doppler, while neglecting the interference term: NRIAcase: same formal result with and with

22. Appendix: average chord length Let : chord length in volume V from on S in the direction with : internal normal ( ) Proportion of chords of length : linked to the corresponding normal cross section: • Average chord length: