PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
CH.VII: NEUTRON SLOWING DOWN
INTRODUCTION
SLOWING DOWN VIA ELASTIC SCATTERING
•
•
•
•
KINEMATICS
SCATTERING LAW
LETHARGY
DIFFERENTIAL CROSS SECTIONS
SLOWING-DOWN EQUATION
• P1 APPROXIMATION
• SLOWING-DOWN DENSITY
• INFINITE HOMOGENEOUS MEDIA
SLOWING DOWN IN HYDROGEN
• HYPOTHESES
• FLUX SHAPE
• SLOWING-DOWN DENSITY SHAPE
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
OTHER MODERATORS
• PLACZEK FUNCTION
• SYNTHETIC SLOWING-DOWN KERNELS
SPATIAL DEPENDENCE
• FERMI’S AGE THEORY
• SLOWING DOWN IN HYDROGEN
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
VII.1 INTRODUCTION
Decrease of the n energy from Efission to Eth due to possibly both
elastic and inelastic collisions
Inelastic collisions: E of the incident n > 1st excitation level of the
nucleus
• 1st excited state for light nuclei: 1 MeV
• 1st excited state for heavy nuclei: 0.1 MeV
 Inelastic collisions mainly with heavy nuclei… but for values
of E > resonance domain
Elastic collisions: not efficient with heavy nuclei
 With light nuclei (moderators)
 Objective of this chapter: study of the n slowing down via
elastic scattering with nuclei of mass A, in the resonance
domain, to feed a multi-group diffusion model (see chap. IV)
in groups of lower energy
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
VII.2 SLOWING DOWN VIA ELASTIC
SCATTERING
KINEMATICS
Absolute coordinates of n
Before collision
v  v
E’
E
Deflection angle:
o  '.
n
vr '
v'
Before collision
After collision
v '  v'  '
G
A G
vG
v
c.o.m. system
After collision
1
vG 
v'  '
A 1
A
vr '  vr  ' 
v'  '
vr  vr r
A 1
vG
Deflection angle:
r  '.r
vr
 Velocity of the c.o.m. conserved
 Velocity modified only in direction in the c.o.m. system
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Minimum energy of a n after a collision
v'
(  ' Ar )
We have v  vG  vr 
A 1
E v 2 A2  2 A r  1
 2 
E ' v'
( A  1) 2
Element

H
0
D2
0.111
 A 1 
Emin  
 E'   E'
 A 1
C
0.716
U238
0.983
2
Thus
Relations between variables
o
1
v
 f (r ) o  v . ' 
1 v'
1
E'
(  ' Ar ). ' 
( A r  1)
v A 1
A 1 E
r  f (E )
( A  1) 2 E A2  1
r 

2 A E'
2A
o  f (E )
o 
A  1 E A 1 E'

2
E'
2
E
r  f (o )
r 
1 2
( o  o A2  1  o2  1)
A
 o 
Ar  1
A2  2 Ar  1
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
SCATTERING LAW
= probability density function (pdf) of the deflection angle
Usually given in the c.o.m. system
Isotropic scattering (c.o.m.) :
p(  r )d r 
1
d r
2
In the lab system: p(r )dr  p(o )do
1 A2  1  2  o2
 2
p( o ) 
 2o
2
2A
A  1  o
For A=1 :

2o
p( o ) | o |  o  
 0
1


A 1
2
(cause vG small)
si o  0
si o  0
 Forward scattering only
Slowing-down kernel (i.e. pdf of the energy of the scattered
n) – isotropic case
K ( E | E' )dE  p(r )dr
1

if E '  E  E '

K ( E | E ' )   (1   ) E '

0
else
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Mean energy loss via elastic collision
 E ' E 
E'

E'
 with E’
(1   )
 with A because

2
(1   ) E '
( E ' E ) K ( E | E ' )dE 
2
2A
( A  1) 2
A
(1-)/2
1
0.5
238 0.0083
LETHARGY
Eo
u  ln
E
Eo : Eréf s.t. u>0 E  Eo = 10 MeV
Elastic slowing-down kernel (isotropic scattering)
e  ( u  u ')
K (u | u' )du  K ( E | E ' )dE 
du
1
with E '  E  E '  u '  u  u ' ln
1

 1

ln

q





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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Mean lethargy increment via elastic collision
  u  u '  
u '  ln
u'
1

(u  u ' ) K (u | u ' )du  1 

1
ln
1 
Independent of u’!
As
( A  1) 2 A  1
  1
ln
,
2A
A 1
=1 for A=1
Mean nb of collisions for a given lethargy increase: n s.t. u=n
Moderator quality
  large + important scattering
 Moderating power: s
 Large moderation power + low absorption
 Moderating ratio: s/a
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Moderator
A


n
s
s/a
H
D
H 2O
D 2O
C
U238
1
2
0
0.111
1
0.725
0.920
0.509
0.158
0.008
14
20
16
29
91
1730
1.35
0.176
0.060
0.003
71
5670
192
0.0092
12
238
0.716
0.983
u s.t. 2 MeV  1 eV
a thermal
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
DIFFERENTIAL CROSS SECTIONS
Link between differential cross section and total scattering
cross section  slowing-down kernel
s (r , u'  u)  s (r , u' ) K (u | u' )
Differential cross section in lethargy and angle:
 s (r , u ' ,  '  u,  )   s (r , u ' ) K (u | u ' )
1
f ( o )
2
Cosinus of the deflection angle: determined by the elastic
collision kinematics
A  1 E A 1 E'
o 

2
E'
2
E
A 1
 o (u  u ' ) 
e
2
u 'u
2
A 1

e
2
u u '
2
f (o )   (o  o (u  u' ))
 Deflection angle determined by the lethargy increment!
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
VII.3 SLOWING-DOWN EQUATION
P1 APPROXIMATION
Comments
Objective of the n slowing down: energy spectrum of the n in
the domain of the elastic collisions
 Input for multi-group diffusion
But no spatial variation of the flux  no current  no diffusion!
 Allowance to be given – even in a simple way – to the
spatial dependence
One speed case: D  1
Here
p( o ) 
1

2A
3 tr
2
A  1  2  o2
A 1  
2
2
o
tr  t   o  s
with
 2o

with <o>0 (mainly if A1)
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Steady-state Boltzmann equation in lethargy
divJ (r , u, )  t (r , u ) (r , u, )


 
u
o
 s (r , u ' , '  u, ) (r , u ' , ' )du' d'  S (r , u, )
d
d
4
(inelastic scattering accounted for in S (outside energy range))
S (r , u,  ) 
1
4


in (r , u '  u ) (r , u ' )du'
1
 (u ) 
4
4
Weak anisotropy 
u
 
o
f
(r , u ' ) (r , u ' ,  ' )du' d 'Q(r , u,  )
 (r , u,  ) 
1
( (r , u )  3.J (r , u ))
4
0th-order momentum
1st-order momentum (S isotropic)
divJ (r , u)  t (r , u ) (r , u)
1
  (r , u )   t (r , u ) J (r , u )
3
u
   s (r , u'  u) (r , u' )du'  S (r , u)
o
u
   s1 (r , u '  u ) J (r , u ' )du'  0
o
with  s1 (r , u'  u)    s (r , u' , '  u, )'.d
12
For a mixture of isotopes:
 s (r , u'  u)    si (r , u' ) Ki (u | u' )
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
i

 s1 (r , u'  u)    si (r , u' ) Ki (u | u' )oi (u  u' )
i
Rem: energy domain of interest: resonance absorptions
 Elastic collisions only
 Inelastic scattering: fast domain  impact on the source
SLOWING-DOWN DENSITY
Angular slowing-down density = nb of n (/volume.t) slowed
down above lethargy u in a given point and direction:
q(r , u, )  
u
o
 
'

u
 s (r , u' , '  u", )du"  (r , u' , ' )du' d'
Slowing-down density:
q( r , u )  
u
o
u

o

 s (r , u'  u" )du"  (r , u' )du'

K (u"| u' )du"  s (r , u' ) (r , u' )du'
 u
 u
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Slowing-down current density:
q1 (r , u)  


u
o

  
u
u
o
'
o

u
 s (r , u' , '  u", )du"  (r , u' , ' )du' d' d

 s1 (r , u'  u" )du" J (r , u' )du'

K (u"| u' )du"  s1 (r , u' ) J (r , u' )du'
 u
 u
Slowing-down density variation:
u
q(r , u )
  s (r , u ) (r , u )    s (r , u '  u ) (r , u ' )du '
o
u
(interpretation?)
 0th-order momentum:
divJ (r , u )   ne (r , u ) (r , u ) 
q(r , u )
 S (r , u )
u
with ne (r , u)  t (r , u)  s (r , u)  a (r , u)  in (r , u) 

 a (r , u )
domain
resonance
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Slowing-down current density variation

u
q1 (r , u )
  s1 (r , u ) J (r , u )    s1 (r , u '  u ) J (r , u ' )du '
o
u
1
q (r , u )
  (r , u )   tr (r , u ) J (r , u )  1
0
3
u
with tr (r , u)  t (r , u)  s1 (r , u)  t (r , u)    oi  si (r , u)
i
Slowing-down equations: summary
Outside the thermal and fast domains:
1
2
3
4
q(r , u )
 S (r , u )
u

  K (u"| u' )du" s (r , u' ) (r , u' )du'
divJ (r , u )   ne (r , u ) (r , u ) 
q(r , u )  
u
o
u
1
q1 (r , u )
  (r , u )   tr (r , u ) J (r , u ) 
0
3
u
u

q1 (r , u)     K (u"| u' )du"  s1 (r , u' ) J (r , u' )du'
o
u
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INFINITE HOMOGENEOUS MEDIA
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Without spatial dependence:
t (u) (u)    s (u'  u) (u' )du'  S (u)
F (u)  t (u) (u)
Collision density:
u
Scattering probability with isotope i:
o
(units?)
ci (u ) 
 si (u )
t (u )
 For an isotropic scattering:
F (u)  
i
1
1  i

u
max(0,u  qi )
e
u 'u
ci (u ' ) F (u' )du'  S (u)
with qi  ln
(interpretation?)
1
i
Rem: F(u) and ci(u) smoother than t(u) and (u)
dq (u )
 S (u )   a (u ) (u )
Slowing-down density:
du
u
Without absorption : q(u)  o  (u)du So for a source S (u)   (u)So
 q(E)/So
= proba not to be absorbed between Esource and E
= resonance escape proba if E = upper bound of thermal E
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
VII.4 SLOWING DOWN IN
HYDROGEN
HYPOTHESES




Infinite media
Absorption in H neglected
Slowing down considered in the resonance domain
Slowing down due to heavy nuclei neglected:
 Elastic: minor contribution
 Inelastic: outside the energy range under study + low proportion of
heavy nuclei
FLUX SHAPE
dF (u )
dS (u )
 (1  c(u )) F (u ) 
 S (u )
du
du
u
F (u)   eu 'u c(u' ) F (u' )du'  S (u)
o
u
F (u )  S (u )   e u '
u
o

(1 c ( u ")) du "
( F (0)  0)
c(u ' ) S (u ' )du '
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
One speed source S (u)  So (u  uo )

F (u )  e

u
uo
 a ( u ')
du '
 t ( u ')
 s (uo )
So
t (uo )
for u > uo
 Superposition of solutions of this type for a general S
Without absorption:
So
 (u) 
t (u)
 (E) 
So
t ( E ) E
With absorption:
 Same behavior for (E) outside resonances (a negligible)
 Reduction after each resonance by a factor   a ( u ') du '
e
res t (u ')
 On the whole resonance domain, flux reduced by

e
u
uo
 a (u ')
du '
t (u ')
e


resu res
 a ( u ')
du '
t ( u ')
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SLOWING-DOWN DENSITY SHAPE
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
u
From the definition : q(u)  o

 u
eu 'u"du" c(u' ) F (u' )du'
u
  eu 'u c(u' ) F (u' )du'  F (u )  S (u )
o
One speed source (uo) and u > uo

q(u)  e

u
uo (1c (u ')) du '
c(uo )So
 Resonance escape proba in u:
 u  a (u ' ) 
q(u )
p(u ) 
 exp  
du' 
q(uo )
 uo t (u ' )

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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
VII.5 OTHER MODERATORS
Reminder:  homogeneous media
F (u)  
i
1
1  i

u
max(0,u  qi )
eu 'u ci (u ' ) F (u' )du'  S (u)
PLACZEK FUNCTION
P(u) = collision density F(u) iff
 One material
 No absorption
 One speed source

P(u)   K (u  u' ) P(u' )du'   (u)
o
with
K (u ) 
1 u
e H (u ) H (q  u )
1
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
1   p 1
Laplace  K ( p) 
(1   )(1  p)
P ( p)  K ( p) P ( p)  1
1
 1  K ( p)  K 2 ( p)  ...
P ( p) 
1  K ( p)
 Inverting term by term, effect of an increasing nb of
collisions
u
eu 'u
P(u ' )du' ?
Solution of P(u )   (u )  
max(0 ,u  q ) 1  
 By intervals of width q
 At the origin: P(u )   (u)
u
exp(
)
1
 1st interval 0 < u < q : P(u) 
1

 Discontinuity in q : 
1
1
u
exp(
)

1
1
nd
(1   
(u  q))
 2 interval q < u < 2q : P(u ) 
1
1
21
 Tauber’s theorem
P  lim pP ( p )
p 0
p
1

p 0 1  K ( p )

 lim
(1-)P(u)
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Asymptotic behavior
Oscillations in the
neighborhood of the origin
=Placzek oscillations
u/q
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
SYNTHETIC SLOWING-DOWN KERNELS
 ordinary diff. eq. for H
 diff. eq. with delay else
 Approximations to simplify this diff. eq.
Integral slowing-down equation
Wigner approximation
Asymptotic behavior of F(u) for an absorbing moderator, with
c(u) cst, for a one speed S:

1c
1 u
Fas (u)  e
c
 Approx. for a slow variation of c(u):
u
 t (u ) o
Fas (u ) 
e
 s (u )

(c1)
 a ( u ')
du '
 t ( u ')
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Slowing-down density:
q(u)  
u
o

u

u

 u
u q
u q
K (u"| u' )du" c(u' ) F (u' )du'
eu 'u "
 u
du" c(u ' ) F (u ' )du'
1
eu '  u  
c(u ' ) F (u ' )du'
1
u ' q
Asymptotic zone (Wigner):
qas (u )  
e
u
qas (u)  e

1
u q

u 'u
u
o
 a (u ')
du '
t (u ')
(u>q)

du' c(u ) Fas (u )
 p(u)
 Resonance escape proba in u
24
Justification of the approximation
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Mean nb of collisions to cross ui where c(ui)  cst: ui/
 Proba to cross without absorption n consecutive intervals
ui/n :
u  (u )
u
u
(1 
i
.
a
i
n t (ui )
p(u)  
i
) n  (1 
i
n
1
(1  c(ui )))n n

 exp(

i
(1  c(ui )))
 1 u

exp(
(1  c(ui )))  exp   (1  c(u' ))du' 

  o

ui
Variation in the approximation
Outside the source domain:
 a (u )
dq as (u )
 a (u )
  a (u ) (u )  
qas (u )
c(u ) Fas (u )  
du
 s (u )
 s (u )
 Age approximation (see below)
1c

1 c u
Rem: compatible with Fas (u)  e
for any c
c
25
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Greuling-Goertzel approximation
d
c(u ' ) F (u ' )  c(u ) F (u )  (u 'u ) (c(u ) F (u ))
we consider
du
d
q
(
u
)


c
(
u
)
F
(
u
)


(c(u ) F (u ))
In the asymptotic zone 
du
1
dq (u )
with    u 2 




c
(
u
)
F
(
u
)


2
du
dq (u )
Yet
  a (u ) (u )
du
Thus  (u ) 
q(u )
 s (u )   a (u )
 a (u )q(u )
dq(u )

du
 s (u )   a (u )
Resonance escape proba
p(u)  e

u
o
a (u ')
du '
 s (u ')a (u ')
Rem: Wigner if 
Age if 0
26
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Generalization: synthetic kernels
Objective: replace the integral slowing-down eq.

F (u)   K (u  u' )c(u' ) F (u' )du' S (u)
o
by an ordinary differential eq. (i.e. without delay)
~
 Synthetic kernel K (u  u' ) close to the initial kernel K (u  u' )
and s.t. approximated solution close enough to F(u)
 Close? Momentums conservation:

 ~
k
M k   K (u)u du   K (u)u k du
o
o
k
m
Choice of the synthetic kernel? Solution of
d
 
Lm (u)   k  
~
 du 
k 0
Lm (u)K (u  u' )  Dn (u) (u  u' ) with
k
n
d 
Dn (u)   k  
 du 
k 0
 approximated diff.eq. for the slowing-down density:
~
~
Lm (u)(F (u)  S (u))  Dn (u)c(u)F (u)
27
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Parameters of the differential operators Lm(u) and Dn(u)?
 Conservation of m+n+1 momentums
1st-order synthetic kernels:
 m=1, n=0  Wigner
 m=0, n=1  age
 m=1, n=1  Greuling – Goertzel
28
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
VII.6 SPATIAL DEPENDENCE
 Slowing down in finite media
FERMI’S AGE THEORY
Use of the P1 equations with
• the age approximation: q(r , u)   (u)s (u) (r , u)
q1 ( r , u )
•
neglected in the current equation
u
1
J (r , u )  
  ( r , u )   D( r , u )   ( r , u )
3tr (r , u )
and  div ( D(r , u )  (r , u ))   a (r , u ) (r , u ) 
q(r , u )
 S (r , u )
u
 homogeneous zone, beyond the sources:
 (r , u )
D(u )
q(r , u )
q(r , u )  a
q(r , u ) 
0
 (u ) s (u )
 (u ) s (u )
u
29
u
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Let  (u)  o
D(u ' )
du' , with  : n age [cm2] !!
 (u ' ) s (u' )
1
q(r , )
q(r , )  2
q ( r , ) 
0
L ( )

Let q(r , )  p( )q~(r , )
with the resonance escape proba p( )  e

Fermi’s equation
u ( )
o
 a ( u ')
du '
  s ( u ')
e


o
1
d '
L2 ( ')
 q~ (r , ) : slowing-down density without absorption
q~ (r , )
~
q ( r ,  ) 

 Equivalent to a time-dependent diffusion equation!
30
Relation lethargy – time ?
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Heavy nuclei  mean lethargy increment low
 low dispersion of the n lethargies
  same moderation
If slowing down identical for all n, u = f(slowing-down time)
With all n with the same lethargy, the diffusion equation at time
t writes (for n emitted at t=0 with u=0):
1  (r , t )
 div ( D(r , t )   (r , t ))   a (r , t ) (r , t )  0
v t
Variation of u / u.t.:
du   s vdt
 Fermi’s equation
Approximation validity
Moderators heavy enough  graphite in practice
31
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Examples of slowing-down kernels
Planar one speed source (Eo)
IC: q~(r ,0)   ( x  xo )
e
~
 q ( r , ) 

| x  xo |2
4
4
Point one speed source (Eo)
~
IC: q (r ,0)   (r  ro )

|r  ro |2
4
e
~
q
(
r
,

)


(4 )3 / 2
Mean square distance to the source:
r2 


o

r 2 q~(r , )4r 2 dr

o
q~(r , )4r 2 dr
 6
 Age = measure of the diffusion during the moderation
32
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Consistent age theory
Same treatment for q1 (r , u) as for q(r , u )
q1 (r , u)  1 (u)s (u) J (r , u)
with
1  
u
o


u
K1 (u"u' )du" du'
K1 (u)  K (u)o (u)
A  1  2 A 1 2
o (u ) 
e 
e
2
2
u
u
33