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

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)
 Effective cross 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 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 !)

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

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

7. 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

8. 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

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: 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. 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 

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

19. 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

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
NR case: without Wigner, with Doppler
Doppler, while neglecting the interference term:
NRIA case: 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: