thermal neutron scattering laws for light and heavy water …...introduction (motivation) •new...

AECL - OFFICIAL USE ONLY / À USAGE EXCLUSIF - EACL

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?

AECL - OFFICIAL USE ONLY / À USAGE EXCLUSIF - EACL 2


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


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


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).


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)


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.


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.


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) .


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


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


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


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)


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


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




Solid-type (phonon-type) gph

(ħ ; T_room) for

D-in-D2O


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




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)

-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


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


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!

•


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


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.

thermal neutron scattering laws for light and heavy water …...introduction (motivation) •new...

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