PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 1 CH.VII: NEUTRON SLOWING DOWN...
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Transcript of PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 1 CH.VII: NEUTRON SLOWING DOWN...
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CH.VII: NEUTRON SLOWING DOWNINTRODUCTION
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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OTHER MODERATORS• PLACZEK FUNCTION • SYNTHETIC SLOWING-DOWN KERNELS
SPATIAL DEPENDENCE• FERMI’S AGE THEORY• SLOWING DOWN IN HYDROGEN
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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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G
KINEMATICS
Absolute coordinates of n c.o.m. system
Velocity of the c.o.m. conserved Velocity modified only in direction in the c.o.m. system
G
Before collision After collision
Deflection angle:
4
VII.2 SLOWING DOWN VIA ELASTIC SCATTERING
Before collision After collision
E’ E
Deflection angle:
''' vv vv
''1
''
vA
Avv rr rrr vv '.o
rr '.
''1
1
v
AvG
nA
'vGv v
Gv
rv
'rv
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Minimum energy of a n after a collision
We have
Thus
Relations between variables
)'(1
'rrG A
A
vvvv
2
2
2
2
)1(
12
''
A
AA
v
v
E
E r
''1
12
min EEA
AE
)( ro f ').'(1
'1'.
1
ro A
A
v
vv
v )1(
'
1
1
rA
E
E
A
)(Efr A
A
E
E
A
Ar 2
1
'2
)1( 22
)(Efo E
EA
E
EAo
'
2
1
'2
1
)( or f )11(1 222 ooor AA
Element H 0
D2 0.111
C 0.716
U238 0.983
12
12
r
ro
AA
A
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SCATTERING LAW
= probability density function (pdf) of the deflection angle
Usually given in the c.o.m. system
Isotropic scattering (c.o.m.) :
In the lab system:
(cause vG small)
For A=1 :
Forward scattering only
Slowing-down kernel (i.e. pdf of the energy of the scattered n) – isotropic case
rrr ddp 2
1)(
oorr dpdp )()(
o
o
oo
A
A
Ap
2
1
21
2
1)(
22
22
2
11
A
00
02||)(
o
ooooo si
sip
rr dpdEEEK )()'|(
else
EEEifEEEK
0
''')1(
1)'|(
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Mean energy loss via elastic collision
with E’ with A because
LETHARGY
Eo : Eréf s.t. u>0 E Eo = 10 MeV
Elastic slowing-down kernel (isotropic scattering)
with
2
')1()'|()'('
'
'
EdEEEKEEEE
E
E
2)1(
2
2
)1(
A
AA (1-)/2
1 0.5
238 0.0083
E
Eu oln
dEEEKduuuK )'|()'|( due uu
1
)'(
1
ln'''' uuuEEE
q
1
ln
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Mean lethargy increment via elastic collision
Independent of u’!
As , =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
1ln
11)'|()'('
1ln'
'
duuuKuuuu
u
u
1
1ln
2
)1(1
2
A
A
A
A
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u s.t. 2 MeV 1 eVa thermal
Moderator A n s s/a
H
D
H2O
D2O
C
U238
1
2
12
238
0
0.111
0.716
0.983
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
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DIFFERENTIAL CROSS SECTIONS
Link between differential cross section and total scattering cross section slowing-down kernel
Differential cross section in lethargy and angle:
Cosinus of the deflection angle: determined by the elastic collision kinematics
Deflection angle determined by the lethargy increment!
)'|()',()',( uuKuruur ss
)(2
1)'|()',(),',',( oss fuuKuruur
E
EA
E
EAo
'
2
1
'2
1
2
'
2
'
2
1
2
1)'(
uuuu
o eA
eA
uu
))'(()( uuf ooo
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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: with
Here with <o>0 (mainly if A1)
sottr tr
D
3
1
o
o
oo
A
A
Ap
2
1
21
2
1)(
22
22
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Steady-state Boltzmann equation in lethargy
(inelastic scattering accounted for in S (outside energy range))
Weak anisotropy
0th-order momentum 1st-order momentum (S isotropic)
)),(.3),((4
1),,( urJurur
),,('')',',(),',',(
),,(),(),,(
4
urSdduuruur
urururJdiv
s
u
o
t
),,('')',',()',()(4
1
')',()',(4
1),,(
4
urQdduururu
duuruururS
f
u
o
in
),(')',()',(
),(),(),(
urSduuruur
urururJdiv
s
u
o
t
0')',()',(
),(),(),(3
1
1
duurJuur
urJurur
s
u
o
t
duuruur ss '.),',',()',(1
dd
with
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For a mixture of isotopes:
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:
Slowing-down density:
)'()'|()',()',(1 uuuuKuruur oiisii
s
)'|()',()',( uuKuruur isii
s
'')',',("),"',',(),,('
dduurduuururq su
u
o
')',(")"',(),( duurduuururq su
u
o
')',()',(")'|"( duururduuuK su
u
o
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Slowing-down current density:
Slowing-down density variation:
(interpretation?)
0th-order momentum:
with
')',()',(),(),(),(
duuruurururu
urqs
u
os
),(),(
),(),(),( urSu
urqurururJdiv ne
),(),(),(),(),( ururururur inastne
ddduurduuururq su
u
o'')',',("),"',',(),(
'1
')',(")"',(1 duurJduuursu
u
o
')',()',(")'|"( 1 duurJurduuuK su
u
o
),( uraresonance
domain
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Slowing-down current density variation
with
Slowing-down equations: summary
Outside the thermal and fast domains:
),(),(
),(),(),( urSu
urqurururJdiv ne
0),(
),(),(),(3
1 1
u
urqurJurur tr
')',()',(")'|"(),( duururduuuKurq su
u
o
')',()',(")'|"(),( 11 duurJurduuuKurq su
u
o
')',()',(),(),(),(
111 duurJuururJuru
urqs
u
os
0
),(),(),(),(
3
1 1
u
urqurJurur tr
),(),(),(),(),( 1 ururururur sioii
tsttr
1
2
3
4
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INFINITE HOMOGENEOUS MEDIA
Without spatial dependence:
Collision density:
Scattering probability with isotope i:
For an isotropic scattering:
with
Rem: F(u) and ci(u) smoother than t(u) and (u)
Slowing-down density:
Without absorption : for a source
q(E)/So = proba not to be absorbed between Esource and E = resonance escape proba if E = upper bound of thermal E
)(')'()'()()( uSduuuuuu s
u
ot
)(')'()'(1
1)( '
),0max(uSduuFuceuF i
uuu
quii i
)()()( uuuF t
)(
)()(
u
uuc
t
sii
iiq
1ln
)()()()(
uuuSdu
udqa
oSuuS )()( o
u
oSduuuq )()(
(interpretation?)
(units?)
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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
)(')'()'()( ' uSduuFuceuF uuu
o )(
)()())(1(
)(uS
du
udSuFuc
du
udF
)0)0(( F')'()'()()(
""))(1(' duuSuceuSuF
duucu
o
u
u
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One speed source
for u > uo
Superposition of solutions of this type for a general S
Without absorption:
With absorption: Same behavior for (E) outside resonances (a negligible)
Reduction after each resonance by a factor
On the whole resonance domain, flux reduced by
)()( oo uuSuS
oot
osdu
u
u
Su
ueuF t
au
ou
)(
)()(
')'(
)'(
)()(
u
Su
t
o
EE
SE
t
o
)()(
')'(
)'(du
u
u
t
a
rese
')'(
)'('
)'(
)'( duu
udu
u
ut
a
resurest
au
ou ee
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SLOWING-DOWN DENSITY SHAPE
From the definition :
One speed source (uo) and u > uo
Resonance escape proba in u:
')'()'(")( "' duuFucdueuq uu
u
u
o
')'()'(' duuFuce uuu
o
)()( uSuF
oo
duuc
Suceuqu
ou )()('))'(1(
')'(
)'(exp
)(
)()( du
u
u
uq
uqup
t
au
uo
o
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VII.5 OTHER MODERATORS
Reminder: homogeneous media
PLACZEK FUNCTION
P(u) = collision density F(u) iff One material No absorption One speed source
with
)(')'()'(1
1)( '
),0max(uSduuFuceuF i
uuu
quii i
)(')'()'()( uduuPuuKuPo
)()(1
1)( uqHuHeuK u
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Laplace
Inverting term by term, effect of an increasing nb of collisions
Solution of ?
By intervals of width q
At the origin:
1st interval 0 < u < q :
Discontinuity in q :
2nd interval q < u < 2q :
)1)(1(
1)(
1
ppK
p
1)()()( pPpKpP
)(1
1)(
pKpP
...)()(1 2 pKpK
')'(1
)()('
),0max(duuP
euuP
uuu
qu
)()( uuP
1
)1
exp()(
u
uP
))(1
1(1
)exp()( 1
11 quuP
u
1
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Asymptotic behavior
Tauber’s theorem
Oscillations in the neighborhood of the origin =Placzek oscillations
1
)(1lim
)(lim
0
0
pK
p
pPpP
p
p
(1-
)P(u
)
u/q
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SYNTHETIC SLOWING-DOWN KERNELS
Integral slowing-down equation ordinary diff. eq. for H
diff. eq. with delay elseApproximations to simplify this diff. eq.
Wigner approximation
Asymptotic behavior of F(u) for an absorbing moderator, with c(u) cst, for a one speed S:
Approx. for a slow variation of c(u):
uc
as ec
uF
1
1)(
')'(
)'(
)(
)()(
duu
u
s
tas
t
au
oeu
uuF
(c1)
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Slowing-down density:
Asymptotic zone (Wigner):
Resonance escape proba in u
')'()'(")'|"()( duuFucduuuKuqu
u
o
')'()'("1
"''duuFucdu
e uuqu
u
u
qu
')'()'(1
'
duuFuce uuu
qu
)()('1
)('
uFucdue
uq as
uuu
quas
)()('
)'(
)'(
upeuqdu
u
u
ast
au
o
(u>q)
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Justification of the approximation
Mean nb of collisions to cross ui where c(ui) cst: ui/Proba to cross without absorption n consecutive intervals
ui/n :
Variation in the approximation
Outside the source domain:
Age approximation (see below)
Rem: compatible with for any c
)))(1(exp()))(1(1())(
)(.1( 1
iinn
iin
it
iai ucu
ucn
u
u
u
n
u
'))'(1(
1exp)))(1(exp()( duucuc
uup
u
oii
i
)()()(
uudu
udqa
as )()()(
)(uFuc
u
uas
s
a
)()(
)(uq
u
uas
s
a
uc
c
as ec
uF
1
1)(
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Greuling-Goertzel approximation
we consider
In the asymptotic zone
with
Yet
Thus
Resonance escape proba
))()(()'()()()'()'( uFucdu
duuuFucuFuc
))()(()()()( uFucdu
duFucuq
2
2
1u
du
udquFuc
)()()(
)()()(
uudu
udqa
)()(
)()(
uu
uqu
as
)()(
)()()(
uu
uqu
du
udq
as
a
')'()'(
)'(
)(du
uu
u
as
au
oeup
Rem: Wigner if Age if 0
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Generalization: synthetic kernels
Objective: replace the integral slowing-down eq.
by an ordinary differential eq. (i.e. without delay)
Synthetic kernel close to the initial kernel and s.t. approximated solution close enough to F(u)
Close? Momentums conservation:
Choice of the synthetic kernel? Solution of
approximated diff.eq. for the slowing-down density:
)(')'()'()'()( uSduuFucuuKuFo
)'(~
uuK )'( uuK
duuuKduuuKM k
o
k
ok )(~
)(
)'()()'(~)( uuuDuuKuL nm
)(~)()())()(
~)(( uFucuDuSuFuL nm
km
kkm du
duL
0
)(
kn
kkn du
duD
0
)( with
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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
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VII.6 SPATIAL DEPENDENCE Slowing down in finite media
FERMI’S AGE THEORY
Use of the P1 equations with
• the age approximation:• neglected in the current equation
homogeneous zone, beyond the sources:
),()()(),( uruuurq s
u
urq
),(1
),(),(
),(),()),(),(( urSu
urqururururDdiv a
),(),(),(),(3
1),( ururDur
ururJ
tr
0),(
),()()(
),(),(
)()(
)(
u
urqurq
uu
ururq
uu
uD
s
a
s
and
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Let , with : n age [cm2] !!
Let
with the resonance escape proba
: slowing-down density without absorption
Equivalent to a time-dependent diffusion equation!
),(~)(),( rqprq
),(~ rq
')'(
1'
)'(
)'(2
)(
)(
d
Ldu
u
uo
s
au
o eep
')'()'(
)'()( du
uu
uDu
s
u
o
),(~
),(~rq
rq
0),(
),()(
1),(
2
rq
rqL
rqFermi’s equation
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31
Relation lethargy – time ?
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):
Variation of u / u.t.:
Fermi’s equation
Approximation validity
Moderators heavy enough graphite in practice
vdtdu s
0),(),()),(),((),(1
trtrtrtrDdiv
t
tr
v a
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
3-2
01
4
32
Examples of slowing-down kernels
Planar one speed source (Eo)
IC:
Point one speed source (Eo)
IC:
Mean square distance to the source:
Age = measure of the diffusion during the moderation
)()0,(~ oxxrq
)()0,(~ orrrq
2/3
4
||
)4(),(~
2
orr
erq
4),(~
4
|| 2oxx
erq
6
4),(~
4),(~
2
22
2
drrrq
drrrqrr
o
o
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
3-2
01
4
33
Consistent age theory
Same treatment for as for
with
),(1 urq ),( urq
),()()(),( 11 urJuuurq s
'")'"(11 duduuuKu
u
o
)()()(1 uuKuK o
22
2
1
2
1)(
uu
o eA
eA
u