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

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)  Effective cross section: with

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  !

Doppler profile for a resonance centered in Eo >> kT ? 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 !)

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

COMPARISON WITH THE NATURAL PROFILE Let: and ( : peak width) Natural profile Bethe-Placzek fcts Doppler profile

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) and

VIII.2 RESONANCE INTEGRAL Absorption rate in a resonance peak: By definition, resonance integral: 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

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: V1 Vo V

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.

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: 

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

NR  NRIA for    ( dilution: I  I) I  if   ([absorbant] ) 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 J(,)  if   (i.e. T ) Fast stabilizing effect linked to the fuel T as  E T   I   p   keff      T 

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 – Cohen method : Intermediate value de  from the slowing-down equation

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 

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

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

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

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

DOPPLER EFFECT IN HETEROGENEOUS MEDIA NR case: without Wigner, with Doppler Doppler, while neglecting the interference term: NRIA case: same formal result with and with

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: