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Thermal Neutron Scattering Laws
for light and heavy water for
Reactor Physics Applications
Dan Roubtsov, Ken Kozier, Björn Becker, Yaron Danon
AECL (CANADA) , RPI (USA)
September 12, 2011
ICTT-22, Portland, Oregon, USA
Introduction (Motivation)
• New neutron Thermal Scattering Laws (TSL) for liquids,
H2O and D2O, are available in
the evaluated nuclear data libraries, such as,
JEFF 3.1 (2005) and ENDF/B-VII.0 (2006).
There are also multi-group libraries developed by
Kyoto University group (Morishima, Edura et al.).
• Then one can ask whether we can now do a better
(more accurate) modeling of the thermal systems with
these new TSL’s.
• And what is actually improved (and what is not) in TSL’s?
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Introduction (Thermal Neutrons)
• Thermal neutrons: neutrons with energies ~ kT (e.g., Tr = 293 K E ~ 0.0253 eV)
Thermal neutrons are subject to up- and down-scatering
(E < E* ~ 10 eV ~ 400 kTr).
typical cut-offs for the thermal neutrons in reactor physics: 10 4 eV < E < 4 eV.
At E > 4-5 eV, epithermal neutrons
Neutron optics: from E < 1 - 5 10 3 eV: cold neutrons,
then very cold, ultra-cold, etc.
At 0.1 - 1 eV < E < 104 - 105 eV, resonance neutrons (low-lying resonances)
• Thermal neutron scattering depends on the isotope and the substance (medium):
chemical/solid state binding effects are important at E < 1 eV.
( E ~ 0.025 eV ~ 1.81 Å, 1/E1/2 )Therefore, for thermal scattering, we use the following “names” for nuclides:
D-in-D2O, H-in-H2O, 16O-in-D2O, 16O-in-H2O (H = 1H, D = 2H),
C-in-Graphite, 238U-in-UO2,
16O-in-UO2, etc.
Introduction ( S( , ) )
• Neutron Thermal Scattering is written in the form of
double differential scattering cross section,
• d2s(E E , )/dE d [ barn per eV per sr]
• and is usually known/used under the name S( , ) data
(for isotope/nuclide X in material Y) in reactor physics community:
d2s(E E , )/dE d =
= ( b/4 kT)(E / E)0.5 exp( /2) S( , ; T)
b is bound scattering cross section of nuclide X,
b/4 = b2 = const for all thermal E of interest, (b (A+1/A) a).
(using Fermi pseudo-potential approx. for V, using Born approximation, etc.)
In S( , ; T), and are dimensionless momentum and energy transfer:
(k k)2 = q2, (E E)
= (E + E 2(E E)0.5 )/AkT and = (E E)/kT = /kT
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Introduction (LEAPR, THERM, etc.)
S( , ; T) are written in Nuclear Data Libraries (in ENDF format, MF = 7)
at fixed temperature nodes Tj .
TSL = Thermal Scattering Law (actually, sub-library) for nuclide X in medium Y
Usually, S( , ; T) data (tables) are generated by NJOY99 using LEAPR module
LEAPR describes a theoretical model of the medium
used to generate S( , ; Tj) on an , grid.
It also writes some additional important parameters (e.g., Teff,j) in MF = 7
that will be used to generate d2s /dE d on an E, E , grid.
THERM module of NJOY99 is used to generate
d2s(E E , ; T)/dE d and the (integral) thermal scattering x-section
s(E; T) using MF = 7 file.
For example, to create a MCNP(X) thermal ACE file at a given T,
one will use LEAPR, THERM, and ACER of NJOY99.
Choice of (E, , E ) is important for accuracy of MC calculations (F4 of MCNP(X)).
Introduction (Incoherent Approximation)
• Coherent and Incoherent components of inelastic d2s:
• d2s(E E , )/dE d = d2
coh(E E , )/dE d +
d2inc(E E , )/dE d
• d2inc b2
inc, d2
inc Sself( , ) (~ self-correlation)
• d2coh b2
coh, d2
coh ( Sself( , ) + Sdistinct( , ) )
(~ self-correlation + pair correlations)
• Incoherent approximation for d2coh: Sdistinct( , ) = 0. Then,
d2s(E E , )/dE d =
= ( b/4 kT)(E / E)0.5 exp( /2) Sself( , ; T), b = b, inc + b, coh
Vineyard (static) approximation: Sdistinct( , ; T) = ( ; T) Sself( , ; T),
where ( ; T) is related to the static structure factor of a liquid,
1+ (q; T) = S(q; T) g(r; T), correlation function of a liquid
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Introduction (Scattering on Molecular Liquids)
• Following Morishima and Aoki (1994), we write the double differential
scattering cross section of
n + H2O / n + D2O in barn/(eV sr) per molecule
using Vineyard approximation:
d2s, inc(E E , )/dE d =
(E / E)0.5 (exp( /2)/kT) n b2inc, n Sself, n( , ; T), n=1,2,3 (e.g., H,H,O)
d2s, coh(E E , )/dE d =
(E / E)0.5 (exp( /2)/kT) n m bcoh, n bcoh, m ( nm + nm(q; T)) Sself, m( , ; T),
n, m = 1,2,3
Need static structure factors of the liquid, Snm(q; T), e.g., for H2O :
1 + HH(q; T) = SHH(q; T), 1 + HO(q; T) = SHO(q; T),
1 + OO(q; T) = SOO(q; T).
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Introduction (H2O)
• n + H2O, thermal neutron scattering on light water
This is a problem with small parameters:
• H: b2inc >> b2
coh ( b2coh,H / b2
inc, H ~ 0.022 )
• O: AO > AH, b2inc, O 0, and b, H >> b, O
b2inc, H >> b2
coh, O ( b2coh, O / b2
inc, H ~ 0.053 )
b2inc, H >> bcoh, Obcoh,H ( bcoh, Obcoh,H / b
2inc, H ~ 0.034 )
Therefore, for n + H2O, the “standard” approximation is
Incoherent Approximation (i.e., nm = 0 for all nuclides n, m).
d2sH-in-H2O(E E , )/dE d =
( b, H/4 kT) (E / E)0.5 exp( /2) Sself, H-in-H2O( , ; T)
In ENDF/B and JEFF, for Oxygen-in-H2O, free gas model at T
Sself, O-in-H2O( , ; T) = SO,free gas( , ; T)
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Introduction (D2O)
• n + D2O, thermal neutron scattering on heavy water
Problem with no small parameters (except AO > AD)
• D: b2inc ~ b2
coh ( b2coh,D / b2
inc, D ~ 2.73 )
• O: AO > AD, b2inc, O 0, but b, D ~ b, O
b2inc, D ~ b2
coh, O ( b2coh, O / b2
inc, D ~ 2.06 )
b2inc, D ~ bcoh, Obcoh, D ( bcoh, Obcoh, D / b
2inc, D ~ 2.37 )
D2O in ENDF/B-VI
Incoherent Approximation, i.e., nm = 0 for all nuclides (n, m = D,D,O)
Then, D-in-D2O and free gas model for Oxygen-in-D2O.
D2O in IKE-2005 JEFF 3.1 ENDF/B-VII.0:
Apply DD(q; T) 0, but disregard DO(q; T) = 0 and OO(q; T) = 0.
Then, new D-in-D2O and free gas model for Oxygen-in-D2O at T.
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Introduction (D2O, Sköld approximation )
n + D2O, thermal neutron scattering on heavy water:
Vineyard approximation for D-in-D2O:
d2sD-in-D2O(E E , )/dE d =
( inc, D/4 kT)(E / E)0.5 exp( /2) Sself, D( , ; T) +
( coh, D/4 kT)(E / E)0.5 exp( /2) SDD(q; T) Sself, D( , ; T)
In fact, in JEFF 3.1 and ENDF/B-VII.0,
Sköld approximation is used for the coherent part of inelastic d2s:
d2sD-in-D2O(E E , )/dE d =
( inc, D/4 kT)(E / E)0.5 exp( /2) Sself, D-in-D2O( , ; T) +
( coh, D/4 kT)(E / E)0.5 exp( /2) SDD(q; T) Sself, D-in-D2O( /SDD(q; T), ; T)
We know s, D, and we need coh, D / s, D 0.732
We also need SDD(q; T) from experiments or (molecular dynamic) simulations.
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Introduction (Egelstaff & Schofield)
• How to calculate Sself( , ; T)?
For inelastic incoherent scattering, apply Gaussian approximation,
(Egelstaff & Schofield 1962) :
exp( /2) Sself, X-in-Y( , ; T) =
(1/2 ħ ) exp(i t/ħ ) exp( q2 WX-in-Y(t)) dt,
The width function W(t):
WX-in-Y(t) =
(ħ2/2MX) 0 d (gX-in-Y( )/ħ ) [coth(ħ /2kT)(1–cos( t)) – isin( t)]
gX-in-Y( ) is a distribution of the normal vibrational modes of nuclide X in mediumY,
0 d gX-in-Y( ) = 1.0, ( [ħ ] = energy), generalized phonon spectrum or DOS.
IF WE KNOW gX-in-Y( ), manage WX-in-Y(t), then we know Sself, X-in-Y( , ; T) .
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Introduction (Theory of Liquids)
• At molecular network level, liquid is characterized by:
self-diffusion process ( tdwell(T), in ps, D(T), in Å2/ps, cluster: n*M )
Hindered translations, hindered rotations (librations), > 0.01 eV
(inter-molecular movements ~ phonons in solids, in meV)
Internal molecular modes at fixed ħ j ~ 0.1-0.4 eV
(but broadened with 10 meV)
gX-in-L( ) = wd gXd( ) + wph gX
ph( ) + wj ( - j), normalized
0 d gX-in-L( ) = 1, weights w are important parameters (w(T)).
In JEFF 3.1 and ENDF/B-VII.0, no self-diffusion for D-in-D2O and
H-in-H2O. Instead
wd gX
d( ) wt ( ) (“free translation“, or gas, approximation at 0),
although Egelstaff & Schoefield diffusion model is available in LEAPR.
Temperature nodes Tjin TSL libraries
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Weights w(T) for H-in-H2O (ENDF/B-VII)
• wt < wph ~ 0.5 and wj ~ 0.16 (j=1,2,3) vs. T
Solid-type (phonon-type) gs(ħ ; T) for H-in-H
2O
• gs(ħ /T) is hindered rotations spectrum (at room temperature)
JEFF 3.1 (= IKE) vs. ENDF/B-VII.0: shift 10 meV
Solid-type (phonon-type) gs(ħ ; T), H-in-H
2O
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gs(ħ /T) is hindered rotations spectrum at T = 450.0 K ,
JEFF 3.1 (=IKE) vs. ENDF/B-VII.0
Solid-type (phonon-type) gph
(ħ , T_room) for H-in-H2O
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D. Altiparmakov (CRL-AECL, 2009): history of gph(ħ )
Frequency distribution for H-in-H2O
Morishima 1994, wd gd( ) + wph gph( ),
at E < 40 meV, hindered translation peak(s)
H-in-H2O for reactor physics applications (1)
H-in-H2O : Teff(B-VII.0) 1270 K ~ Teff(JEFF 3.1) 1300 K ( ~ 0.11 eV)
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CANDU 6 type bundle, but
H2O cooled / D
2O moderated, L.P. ~ 28.6 cm, RPT ~ 5.2 cm
D2OModerator,
s,tr 2 - 2.5 cm
UO2 / MOXfuel pins
R ~ 0.6 cm
H2O coolant,
s,tr 0.2 - 0.5 cm
ZED-2 reactor in CRL (AECL):
D2O moderated, Al vessel, Graphite reflector
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Critical lattices of U-nat/LEU fuel in channels with D2O /
H2O / air coolant.
IKE (= JEFF 3.1) and ENDF/B-VII.0 models
for H-in-H2O are (slightly) different, so...
IF H-1 TSL worth, i.e.,
k-effective( Free Gas Model H ) – k-effective( TSL H-in-H2O ),
is large enough, one could examine different models of H-in-H2O and validate them (?).
We have TSL worth for H-in-H2O ~ 10 - 16 mk in the coolant (10 mk = 1000 pcm)
IKE (JEFF 3.1) and ENDF/B-VII.0 models
for H-in-H2O are (slightly) different, and
• Then, k-effective (IKE H-in-H2O) – k-effective (B-VII.0 H-in-H2O) ~ 1 – 3 mk,
(1 mk = 100 pcm)
• and ENDF/B-VII.0 H-in-H2O seems to give better results for critical cores!
(i.e., k-effective bias, k-eff – 1.0, is better with ENDF/B-VII.0 H-in-H2O.)
Librations (~ phonon-type) gph
(ħ ; T) for D-in-D2O
• gph(ħ ) = libration spectrum (ENDF/B-VII.0 = JEFF 3.1)
(weights wt < wph ~ 0.45 and wj ~ 0.16 do not depend on T)
0
1
2
3
4
5
6
7
8
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17
rho
( o
me
ga
) [
1 / e
V ]
omega [ eV ]
Continuous part of frequency spectrum for H-2 in D2O, ENDF/B-VII.0 thermal data
IKE-Stuttgard spectrum, T = 293.6 K
IKE-Stuttgard spectrum, T = 350.0 K
IKE-Stuttgard spectrum, T = 400.0 K
IKE-Stuttgard spectrum, T = 450.0 K
Solid-type (phonon-type) gph
(ħ ; T_room) for
D-in-D2O
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D. Altiparmakov (CRL-AECL, 2009): history of gph(ħ )
Frequency distribution for D-in-D2O
• Morishima 1994, wd gd( ) + wph gph( )
• at E < 30 meV, hindered translation peak
Static structure factor for D-in-D2O
SDD(q; T) = static structure factor is used in IKE model;
it is based on simulations (JEFF 3.1 ENDF/B-VII.0).
0
0.5
1
1.5
2
2.5
3
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
S(
ka
pp
a )
kappa [ 1/A ]
Stucture factor for H-2 in D2O, ENDF/B-VII.0 thermal scattering
Skold model is used in the H-2 leapr inputs
IKE-Stuttgard S, T = 293.6 K
IKE-Stuttgard S, T = 350.0 K
IKE-Stuttgard S, T = 400.0 K
IKE-Stuttgard S, T = 450.0 K
D-in-D2O for reactor physics aplications (1)
D-in-D2O : Teff(B-VII.0) 1010 K ~ Teff(B-VI) 940 K ( ~ 0.08 - 0.09 eV)
D-in-D2O for reactor physics aplications (2)
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-bar(E) in D-in-D2O and H-in-H
2O
D-in-D2O Teff ~ 1030 K < H-in-H2O Teff ~ 1300 K
CANDU 6 bundles: D2O cooled / D
2O moderated
IF Deuterim TSL worth, i.e.,
k-effective( Free Gas Model D ) –
k-effective( D-in-D2O ),
is large enough, one could examine
differences between different models
of D-in-D2O.
We have D TSL worth ~ 1 - 6 mk for D-in-D2O in ZED-2 models,
and we did not notice any essential differences between
ENDF/B-VI and ENDF/B-VII.0 models of D-in-D2O (yet...).
(Differences in k-effective are < or ~ 0.1 mk, (1 mk = 100 pcm ).)
Moderator, D2O
Coolant, D2OFuel pin UO2
Calandria tube
Gap
Pressure tube
D-in-D2O for reactor physics aplications (3)
TSL worth for D-in-D2O is not negligible (ranges from -0.5 to +6 mk).
It depends on the lattice pitch (hardness of n-spectrum), type of coolant,
etc. ...
B-VI TSL model vs. B-VII.0 one for D-in-D2O (1)
• We are interested in the thermal energies, 10 2 -10 1 eV.
B-VI vs. B-VII.0 for D-in-D2O (2)
B-VI vs. B-VII.0 for D-in-D2O, at T_room (3)
• Mattes and Keinert, IKE-2005
• We believe that some improvements in D2O model (LEAPR) could help
at 10 2 -10 1 eV; new experiments with 10 deg C “T grid” (?)
Simulations of time-of-flight (TOF)
measurement set-up
• Neutron leakage experiment
• Time of flight measurement
(15 m) at 90 deg.
• Simulations (MCNP)
–Neutron source: approx.
evaporation spectrum
–No moderation within
target (Ta)
–10cm x 10cm x 10cm
water cube
–F5 tally
• Simulation in time and the
converted to energy
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Vacuum
Electron
Beam
Neutron Source
(Ta Target)
Neutron
Detector
TOF
(15 m)
Moderator Slab
(H2O, D
2O or
CH2)
TOF simulations: Results (1)
• Leakage of a 10 x 10 x 10 cm
cube of water
• Evaporation neutron spectrum
• 15 m TOF
• Flux converted from time to
energy dependent
• About 25 % difference between
S(α,β) H-in-H2O and FG
scattering observed
• The difference between S(α,β)
tables too small to measure with
this setup --> full scattering
experiment necessary!
•
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1E-3 0.01 0.1 1 100
1
2
3
4
Ne
utr
on
Flu
x a
t D
ete
cto
r (E
*)
Energy [eV]
ENDF/B 7, No S( )
ENDF/B 7, S( ): H-in-H2O (ENDF/B 7 )
ENDF/B 7, S( ): H-in-H2O (IKE)
TOF simulation results: H-in-H2O (2)
• Leakage of a 5x5x5 cm cube of
water
• Evaporation neutron spectrum
• 15 m TOF
• Flux converted from time to energy
dependent
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1E-3 0.01 0.1 1 100.00
0.05
0.10
0.15
0.20
0.25
0.30
Ne
utr
on
Flu
x a
t D
ete
cto
r (E
*)
Energy [eV]
ENDF/B 7, No S( )
ENDF/B 7, S( ): H-in-H2O (ENDF/B 7 )
ENDF/B 7, S( ): H-in-H2O (IKE)
Conclusion
• TSL for H-in-H2O: TSL worth ~ 10 – 15 mk
it seems that ENDF/B-VII.0 improvements of the IKE-2005 (= JEFF 3.1) model work in the right directions.
One can elaborate more / improve the generalized frequency spectrum,
e.g., add gd( ) and gph( ) for H2O at low ħ (hindered translation),
It seems that this TSL is good enough for the reactor physics applications,
…but the model improvements are always worth doing…
TSL for D-in-D2O: TSL worth ~ 1 – 6 mk (CANDU type bundles)
IKE-2005 (= JEFF 3.1) model improved the accuracy: e.g., we have a “coherence dip“ at cold neutron energies and D2O at T ~ T_room.
The discrepancy at the thermal neutron energies is not fully addressed yet.
Some improvements in LEAPR model and LEAPR capabilities are desirable,
e.g., add gd( ) and improve gph( ) for D2O
have both DD(q; T) and DO(q; T) structure corrections in the coherent part of d2s,
16O-in-D2O (Teff) and new scattering experiments.
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