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
1
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 nuclei!
 Reaction rate:
R   n(v )dv  N (| v  V |). | v  V | P (V )dV
v
where v,V
V
: absolute velocities of the n and nucleus, resp.
But P and : f (scalar v)
 Effective cross section:
v eff (v)    (| v  V |). | v  V | P (V )dV
V

R   N eff (v) (v)dv with  (v) 
o
2
vn
(
v
)
v
d

4
2
Particular cases
Let
vr  v  V
1.   c / vr
 eff (v) 
1
c
c
.
|
v

V
|
P
(
V
)
d
V

v V | v  V |
v
 Profile in the relative v unchanged in the absolute v
2.  slowly variable and velocity above the thermal domain
vr  v
 eff (v) 
1
 (v)vP(V )dV   (v)

vV
 Conservation of the relative profiles outside the resonances
3. Energy of the n low compared to the thermal zone
1
c ste
 eff (v)    (V )VP (V )dV 
vV
v
vr  V
 indep. of  !
 Effect of the thermal motion on measurements of  at low E
3
DOPPLER EFFECT
Rem: v eff (v) = convolution of | v | . (| v |) and P (V )  widening
of the resonance peak
Doppler profile for a resonance centered in Eo >> kT ?
(Eo: energy of the
relative motion !)
Maxwellian spectrum for the thermal motion:
3/ 2
 M 
P(V )dV  
 .e
 2kT 

MV 2
2 kT
dV
3/ 2
 M  1
 .  v r  ( vr ) e
 2kT  v vr
 eff (v)  
Effective cross section:
1/ 2
 M  1
 eff (v)  
 . 2
 2kT  v


o
1/ 2
M ( v  vr )
  M ( v  vr )

2
vr  (vr ) e 2 kT  e 2 kT


 M  1
 . 2
 2kT  v
 eff (v)  
2


o
vr2 (vr )e

M ( v  vr ) 2
2 kT
2
M |v  vr |2

2 kT
dvr

dvr


dvr
4
v 2  vr
v  vr 
2vr
Approximation:
Let:

mM
M m
2
: reduced mass of the n-nucleus system
vr 2
~ v 2
E
, Er 
2
2
4kTEo
: Doppler width of the peak

M
~
 eff ( E ) 

Eo
. ~
 E
1
1
 

o



Er  ( Er ) e
o
 ( Er )e

~
( E  Er ) 2
2 Er
E
o
dEr
~
( E  Er ) 2
2
dEr
5
COMPARISON WITH THE NATURAL PROFILE
~

E r  Eo
E  Eo
and  
x
, y

/2
/2
Let:
 eff ( x) 
Natural profile
a  o



 ( y)e
 2 ( x y )2
4
dy
Bethe-Placzek fcts
Doppler profile

1
 1 y2
 a eff   o
 1
s  o n
  si   pa
 1 y2
t  o

2 

( : peak width)
 ( , x) 

2 





1
e
1 y2
2
( x y )2
4
dy
1
  si   pa
2
1 y
 

 si   o pa g J n 


1/ 2
 pa  4R2
2y
1 y2
 s eff   o


 ( , x)
n
 ( , x)   si eff   pa

 t eff   o ( , x)   si eff   pa
 ( , x) 

2
 


2 y 
e
2
1 y
2
( x y )2
4
 

  o pa g J n   ( , x)


1/ 2
dy
 si eff
 pa eff  4R2
6
Properties of the Bethe – Placzek functions
1.
 (low to)
1
 ( , x) 
1 x2

Natural profiles
2.
0 (high to)
and  ( , x) 
 ( , x) 
3.





2
 .e

2x
1 x2
 2 x2
4
 ( , x)dx  
 Widening of the peak, but conservation of the total surface
below the resonance peak (in this approximation)
7
VIII.2 RESONANCE INTEGRAL
Absorption rate in a resonance peak: Ra 

a
(u ) (u )du
Rés
 (u)
du
By definition, resonance integral: I    a (u)
as
Rés
Ra  NIas
(as: asymptotic flux, i.e. without resonance)
 I : equivalent cross section
Flux depression in the resonance but slowing-down density +/cst on the u of 1 unique resonance
 If absorption weak or = :
q(u)  t (u) (u)  q
Before the resonance: q   pas because t   p  c ste(p : scattering
of potential)
I
  a (u)
Rés
p
t (u)
du
and
Ra  q
NI
 p
8
Resonance escape proba
 NI 
q  Ra

p
 exp 
  
q
p 

 N i I i 

p  exp 




p


For a set of isolated resonances:
Homogeneous mix
Ex: moderator m and absorbing heavy nuclei a
 p  a pa  mm
Heterogeneous mix
V1
Vo
Ex: fuel cell
V
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:  p 
 Resonance escape proba
 o  poVo  1 p1V1
V
 NI Vo 

p  exp 
  V 
p


9
VIII.3 RESONANCE INTEGRAL –
HOMOGENEOUS THERMAL REACTORS
INFINITE DILUTION
Very few absorbing atoms  (u) = as(u)
dE

I     a eff ( E )

E
2Eo
Rés

a eff
( y)dy 
Rés

2Eo
 o
(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.
a  N a
t  N t  m
 p  N pa  m
10
Microscopic cross sections per absorbing atom
m
m 
N

 tot  t   t   m
N
p
p 
  pa   m
N
NR approximation (narrow resonance)
Narrow resonance s.t.    E'E m and    E'E a
i.e., in terms of moderation, qi >> ures:
F (u)  
i

u
u  qi
eu '  u
 si (u' ) (u' )du' 
1  i

i
u  qi
By definition: F (u)  t (u) (u)
p
 (u)
    (u)
as
tot
I
u
NR
eu '  u
du'  pias   pas
1  i
 a (u) p
 
du
 tot (u)
Rés
11
NRIA approximation (narrow resonance, infinite mass
absorber)
Narrow resonance s.t.    E'E m but
 E'E a  0
(resonance large enough to undergo several collisions with the
absorbant  wide resonance, WR)
 K (u)   (u) for the absorbant
Thus
F (u)  t (u) (u)  sa (u) (u)  mas
I NRIA 
 a (u ) m
Rés  a (u)   m du
Natural profile  I 
*
with

NR
 pa   m

o
 o 
*
2 Eo
 * 1
and 
NRIA
m 

 o 
12
Remarks
 NR  NRIA for    ( dilution: I  I)
 I  if   ([absorbant] )
 Resonance self-protection: depression of the flux reduces
as
the value of I

Doppler profile  I 
*
with
 o 
Eo
E
 * J ( ,  * )

 ( , x)
1   ( , x)

dx
J ( ,  )  
dx o
  ( , x)
2    ( , x)
Remarks
J(,)  if   (i.e. T )
 Fast stabilizing effect linked to the fuel T
T   I   p   keff      T 
13
14
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 ?
 m   pa


We can write
with =0 (NRIA),1 (NR)
o
  n
 m   pa
 (u)

as  a   m   pa
 Goldstein – Cohen method :
Intermediate value de  from
 (u) u eu 'u
 (u' )
the slowing-down equation
t (u)

 so (u' )
du' m
as
u  qo
1  o
as
15
VIII.4 RESONANCE INTEGRAL –
HETEROGENEOUS THERMAL REACTORS
GEOMETRIC SELF-PROTECTION
Outside resonances (see above):
Asymptotic flux spatially uniform
q
as 
 p
with
In the resonances:
 p 
 o  poVo  1 p1V1
V
V1
Vo
V
(Rem: fuel partially moderating)
o po  a  pa  mm
Strong depression of the flux in Vo  I 
 Geometric self-protection of the resonance
 Justification of the use of heterogeneous reactors (see notes)
16
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 ?
u
Voto (u)o (u)  (1  Po ) Vo u q Ko (u  u' )so (u' )o (u' )du'
o
u
 P1 V1 u q K1 (u  u' ) s1 (u' )1 (u' )du'
1
NR approximation
qo, q1 >> 
u
Voto (u)o (u)  (1  Po )Vo u q Ko (u  u' )du'  poas
o
u
 P1V1 
u q1
K1 (u  u' )du'  p1as
Rem: Pk = leakage proba without collision
17
Reminder chap.II
Relation between Po and P1
p1V1P1 = to(u)VoPo
18
Thus
 po
 po
to (u)   po
o (u)
 (1  Po )
 Po 
 Po
to (u)
to (u)
as
to (u )
1
Wigner approximation for Po : Po 
1   to 
with l : average chord length in the fuel (see appendix)
o (u)  po   1 



as
1  to 
*
p
*
t
I NR
*m  m  1 / 
 

N
N
*t
*
 t    t   m*
N
*

 *p  p   pa   m*
N
*
m
with
 a (u ) *p
 
du
*
 t (u )   m
Rés
NRIA approximation

u
Vo (a (u)  m ) (u)  (1  Po )Vo 
u qm
Km (u  u' )du' mas  P1V1 p1as
since the absorbant does not moderate
19
Thus
m
 a (u )
 (u )

 Po
as
 a (u )   m
 a (u )   m
Pk : leakage proba, with or without collision
s
 If Pc = capture proba for 1 n emitted …, then Po  1  Pc  Pc Po
t
t 
1
Pc 
Wigner approx:
Po 
1  ( a   m ) 
1  t 
 m*
 (u)

as  a (u)   m*
and I
NRIA
 a (u) m*
 
du
*
 t (u)   m
Rés
 INR, INRIA formally similar to the homogeneous case
 Equivalence theorems
20
DOPPLER EFFECT IN HETEROGENEOUS MEDIA
NR case: without Wigner, with Doppler
 a (u) p
t (u)   p
NR
I  
du   Po a (u)
du
Rés
t (u)
 a (u) p
t (u)
Rés
 t (u)   p
 
du   Po (t (u) ) a (u)
du
 t (u)   m
 t (u)   m
Rés
Rés
Doppler, while neglecting the interference term:
t   N ( o ( , x)   p )   N p (1 
 ( , x)
 t (1 
)
 ( , x)
)

with
 pa   m

o

I
NR
 o
 o
( ( , x))2

J ( ,  ) 
Po (t ( x)l )
dx



Eo
2Eo
 ( , x)  
 o 
 o 

1 
( ( , x))2

J ( ,  ) 
L(t , ,  )  L(t , ,  )   Po (t ( ( , x)  1))
Eo
Eo


NRIA case: same formal result with  
2

dx
 ( , x)   
m 
and t  N m 
 o 
21
Appendix: average chord length
Let (rs , ) : chord length in volume V from rs
direction 
on S in the
V   ni . ( rs ,  ) dS
S
with ni : internal normal ( (rs , )  0 if ni .  0 )
Proportion of chords of length  : linked to the corresponding
normal cross section:

 ()d 
 
ni .dSd
ni . dS
 ( rs ,  )  
  ( rs ,  )  
  n .dSd
i
 S

  (r , )n .dSd
 Average chord length:    o  ()d  
s
S
i
  n .dSd
i
 o  a    Po

4V
S
 S
22