Embed Size (px)
Citation preview
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
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• NR AND NRIA APPROXIMATIONS• DOPPLER EFFECT
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
EFFECTIVE CROSS SECTIONSCross 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:
2
VIII.1 RESONANCE CROSS SECTIONS
VdVPVvVvNvdvnRVv
)(|||).(|)(
VdVPVvVvvvV
eff )(|||).(|)(
dvvvNR effo)()(
dvvvnv 2
4
)()(
Vv ,
with
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
3
Particular casesLet
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
v
cVdVPVv
Vv
c
vv
V
eff
)(||.||
1)(
Vvvr
vvr
)()()(1
)( vVdVvPvv
vV
eff
rvc /
Vvr
v
cVdVVPV
vv
ste
V
eff )()(1
)( indep. of !
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
4
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:
|)(|.|| vv )(VP)(vv eff
VdekT
MVdVP kT
MV
2
2/3 2
.2
)(
rkT
vvM
rr
v
eff vdevvvkT
Mv
r
r
2
||2/3 2
)(1
.2
)(
rkT
vvM
rroeff dvevvvkT
Mv
r
2
)(2
2
2/1 2
)(1
.2
)(
rkT
vvM
kT
vvM
rroeff dveevvvkT
Mv
rr
2
)(
2
)(2
2
2/1 22
)(1
.2
)(
(Eo: energy of therelative motion !)
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
5
Approximation:
Let:
r
rr v
vvvv
2
22
,2
~ 2vE
2
2r
r
vE
M
EkT o4
mM
mM
r
EE
rro
oeff dEeEE
E
EE oE
rEr
2
2)~
(
)(~.1
)~
(
r
EE
rodEeE
r2
2)~
(
)(1
: reduced mass of the n-nucleus system
: Doppler width of the peak
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
6
COMPARISON WITH THE NATURAL PROFILE
Let:
21
1
yoa
pasin
os y
21
1
pasiot y
21
1
2
2/1
1
2
y
yg n
Jpaosi
24 Rpa
2/
or EEy
dyeyxyx
eff4
)( 22
)(2
)(
,2/
~
oEEx
dyey
xyx
4
)(
2
22
1
1
2),(
dyey
yx
yx
4
)(
2
22
1
2
2),(
Natural profile Bethe-Placzek fcts Doppler profile
and ( : peak width)
),( xoeffa
paeffsin
oeffs x
),(
paeffsioefft x ),(
),(2/1
xg nJpaoeffsi
24 Reffpa
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
7
Properties of the Bethe – Placzek functions
1. (low to)
Natural profiles
2. 0 (high to)
3.
Widening of the peak, but conservation of the total surface below the resonance peak (in this approximation)
dxx),(
21
1),(
xx
21
2),(
x
xx
4
22
.2
),(x
ex
and
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
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
duuuR a
Rés
a )()(
duu
uIas
a
Rés )(
)(
asa NIR
quuuq t )()()(
aspq
duu
uIt
pa
Rés )()(
p
a
NIqR
stept c
and
(as: asymptotic flux, i.e. without resonance)
I : equivalent cross section
(p : scattering of potential)
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
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:
Resonance escape proba
mmpaap
p
a NI
q
Rqp
exp
p
iiIN
p
exp
V
VV popoop
111
V
VNIp o
pexp
Vo
V1
V
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
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.
E
dEEI effa
Rés
)( o
oeffa
Réso Edyy
E 2)(
2
aa N
mtt N mpap N
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
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:
mEE '
Nm
m
mtt
tot N
mpap
p N
aEE '
)()()( uuuF t
')'()'(1
)('
duuue
uF sii
uuu
qui i
aspii
uuu
qui
due
i
'1
'
asp
)(
)(
u
u
tot
p
as
duu
uI
tot
pa
Rés
NR
)(
)(
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
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
mEE ' 0' aEE
asmsat uuuuuF )()()()()(
duu
uI
ma
ma
Rés
NRIA
)(
)(
)()( uuK
12 *
**
o
o
EI
o
mpaNR
o
mNRIA
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
13
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
Remarks
J(,) if (i.e. T ) Fast stabilizing effect linked to the fuel T
),( *** J
EI
o
o
dxx
xJ
),(
),(
2
1),(
dx
x
xo ),(
),(
E
as
T I p keff T
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
14
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
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
no
pam
pama
pam
as
u
)(
mas
soo
uuu
quas
t duu
ueu
uo
')'(
)'(1
)()(
'
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
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)
p
as
q
Vo
V1
VV
VV popoop
111
I
mmpaapoo
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
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
)()( uuV otoo ')'()'()'( duuuuuKV osoo
u
quoo
')'()'()'( 11111
duuuuuKV s
u
qu
)1( oP
)()( uuV otoo
1P
aspoo
u
quoo duuuKVPo
')'()1(
asp
u
quduuuKVP 1111 ')'(
1
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
18
Reminder chap.II
Relation between Po and P1p1V1P1 = to(u)VoPo
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
19
Thus
Wigner approximation for Po :
with l : average chord length in the fuel (see appendix)
NRIA approximation
since the absorbant does not moderate
oto
poo
as
o Pu
Pu
)(
)1()(
tooP
1
1
to
po
as
o u
1
1)(
*
*
t
p
NNmm
m
/1**
**
*mt
tt N
**
*mpa
pp N
du
u
uI
mt
pa
Rés
NR*
*
)(
)(
aspasmm
u
quoomao VPduuuKVPuuVm
111')'()1()())((
)(
)(
)( u
uP
u to
potoo
to
po
with
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
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 caseEquivalence theorems
ma
ao
ma
m
as u
uP
u
u
)(
)(
)(
)(
oct
sco PPPP
1
t
tcP
1 )(1
1
maoP
*
*
)(
)(
ma
m
as u
u
du
u
uI
mt
ma
Rés
NRIA*
*
)(
)(
and
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
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
duu
uuPdu
u
uI
t
ptao
Rést
pa
Rés
NR
)(
)()(
)(
)(
duu
uuuPdu
u
u
mt
ptato
Résmt
pa
Rés
)(
)()())((
)(
)(
dxx
xlxP
EJ
EI to
o
o
o
oNR
),(
)),(())((
2),(
2
)),(
1()),(( x
NxN ppot o
mpa
)),(
1( x
t
),,(),(
tL
EJ
E o
o
o
o
dx
x
xxtPtL o
),(
)),(())1),(((
2
1),,(
2
o
m mNt
with
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
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:
),( sr
dSrnV si
S
),(.
dSni
rs
.),(
dSdn
dSdn
di
S
i
rs
.
.
)( ),(
S
V
dSdn
dSdnr
di
S
is
S
o
4
.
.),(
)(
0.0),( is nifrin
sr
oao P
