Implementation of

Atomistic Simulation Techniques in Thermal Neutron Scattering

Data Generation

Ayman I. Hawari

Nuclear Reactor Program Department of Nuclear Engineering

North Carolina State University Raleigh, North Carolina, USA

OECD/NEA Kick-off Meeting: WPEC SG42 Thermal Scattering Kernel S(,): Measurement, Evaluation and Application

May 18 19, 2015 Paris, France

Acknowledgement The many graduate students and postdocs at

North Carolina State University

Acknowledgement Collaboration with LLNL and Bettis labs Current funding by the NCSP and NR programs

and past funding by DOE NE through NERI and NEUP programs

Motivation

5

Example

Vision Establish a predictive approach for generating the

needed data (cross sections) to describe the energy exchange of thermal neutrons in matter

Various applications:

Nuclear criticality safety Nuclear reactor design Neutron beam spectral shaping (i.e., filtering) Neutron source (cold, ultracold, etc.) characterization

{ })()( ,S,Skk

41

Eddd

sincohcoh

2 +

=

)()()( ,S,S,S ds

+=

is composed of two parts )( ,S The scattering law

Using Born approximation, it can be shown that the double differential scattering cross section has the form

Van Hoves formulation

( ) ( ) ( )1, ,2

i r tS G r t e drdt

=

where G(r,t) is the dynamic pair correlation function and is expressible in terms of the atomic positions.

Thermal Neutron Scattering

Correlation Functions Fourier transforms of correlation functions in space and

time reveal the energy and momentum states available within the system

radius []0 2 4 6 8 10

Log[

G(r

)] [a

rb.]

time [fs]0 1000 2000 3000 4000 5000

VA

CF

[arb

.]

-1.0

-0.5

0.0

0.5

1.0

Frequency [THz]0.0 0.5 1.0 1.5 2.0 2.5

DO

S [a

rb.]

q-vector [-1]

0.0 0.1 0.2 0.3 0.4 0.5 0.6

I(k)

[arb

.]

(1)

Methods Several approaches can be used to extract the

fundamental information for calculating the scattering law and eventually the cross sections

Empirical atomic force analysis combined with dynamical

matrix calculations Basis of current ENDF/B libraries

Ab initio Quantum (DFT) methods combined with dynamical

matrix calculations Molecular Dynamics (ab initio MD or classical MD) methods

combined with correlation function analysis

e.g., NJOY

e.g., NJOY

Lattice Dynamics

Hellmann-Feynman Forces (HF)

Ab Initio DFT

()

One Phonon

)(qj)(qe jd

),(1 dS),( sS

Scattering Cross Section ( )

( ),S ( ),sS

Velocity autocorrelation

MD Simulation

Particle density autocorrelation

Time Transform

( ){ }jR t

( )

( ){ }jv t

Time Transform

( ),I t

Build Atomistic Model

Materials Studied at NCSU Graphite, Beryllium (PHYSOR 2004 & 2008)

Treatment of nuclear graphite (porous system) Including coherent inelastic for both materials

Silicon dioxide (New, contributed to NNDC/ENDF, PHYSOR 2008) Support criticality safety analysis

Silicon carbide (New, contributed to NNDC/ENDF) Support advanced fuel cycle applications (e.g., FCM fuels)

Thorium hydride, uranium-zirconium hydride, calcium hydride

(PHYSOR 2004)

Sapphire and bismuth (PHYSOR 2006) Thermal neutron filters

Solid methane (predictive analysis AccApp 2011) Cold neutron moderator Captured phase I to II transformation upon cooling below 22 K

Lucite (C5O2H8) and polyethylene (C2H4) Of interest as moderators

Graphite

Hexagonal Structure 4 atoms per unit cell a=b=2.46 c=6.7

Ideal graphite consists of planes (sheets) of carbon atoms arranged in a hexagonal lattice. Covalent bonding exits between intraplaner atoms, while the interplaner bonding is of the weak Van der Waals type. The planes are stacked in an abab sequence.

Graphite 1

T = 300 K

Energy (eV)10-5 10-4 10-3 10-2 10-1 100

Cro

ss S

ectio

n (b

)

10-1

100

101

102

ENDF/B-VII data - calculated

T = 300 K

Energy (eV)10-5 10-4 10-3 10-2 10-1 100

Cro

ss S

ectio

n (b

)

10-1

100

101

102

ENDF/B-VII data - calculatedIdeal graphite - measuredBNL-325 - measuredNCSU ideal graphite - measuredNCSU reactor-grade graphite - measured

Graphite 2

- particle density autocorrelation ( ),S Scattering law is computed directly from the atomic positions as:

( ) ( ) ( )', '

1, exp 0 expcl j jj j

I t i R i R tN

=

( ) ( ) ( )1, , exp2

clS I t i t dt

= where the positions are known at discrete time steps and the Fourier transforms are taken in discrete form.

Detailed balance

Response of scattering system to neutron interaction

Graphite Scattering Law

T = 300 K

Energy (eV)10-5 10-4 10-3 10-2 10-1 100

Cro

ss S

ectio

n (b

)

10-1

100

101

102

ENDF/B-VII data - calculatedIdeal graphite - measuredBNL-325 - measuredNCSU ideal graphite - measuredNCSU reactor-grade graphite - measured

Graphite 2

Graphite 3

T = 300 K

Energy (eV)10-5 10-4 10-3 10-2 10-1 100

Cro

ss S

ectio

n (b

)

10-1

100

101

102

Ideal graphite - calculatedENDF/B-VII data - calculatedIdeal graphite - measuredBNL-325 - measuredNCSU ideal graphite - measuredNCSU reactor-grade graphite - measured

Graphite Types

Ideal Graphite

Density = 2.25 g/cm3

Nuclear Graphite

Density = 1.5 1.8 g/cm3

MD Models

8000 atoms 30% porosity NVT ensemble T = 300 K

MD Models

Energy (eV)

0.00 0.05 0.10 0.15 0.20 0.25

DO

S (1

/eV

)

0

10

20

3010% porous graphite30% poorous graphiteIdeal graphite

Graphite 3

T = 300 K

Energy (eV)10-5 10-4 10-3 10-2 10-1 100

Cro

ss S

ectio

n (b

)

10-1

100

101

102

Ideal graphite - calculatedENDF/B-VII data - calculatedIdeal graphite - measuredBNL-325 - measuredNCSU ideal graphite - measuredNCSU reactor-grade graphite - measured

Graphite 4

T = 300 K

Energy (eV)10-5 10-4 10-3 10-2 10-1 100

Cro

ss S

ectio

n (b

)

10-1

100

101

102

30% porous graphite - calculated10% porous graphite - calculatedIdeal graphite - calculatedENDF/B-VII data - calculatedIdeal graphite - measuredBNL-325 - measuredNCSU ideal graphite - measuredNCSU reactor-grade graphite - measured

AIMD Model of Be HCP (P63/mmc)

a=2.2856 (2.27 AIMD) c=3.5832 (3.55 AIMD)

VASP

GGA-PAW 3x3x3 k-mesh 350eV Plane-wave cut-off

AIMD Model of Be VASP 5x5x5 super-cell (250 atoms) Temperature of 300K under NVE Equilibrated with velocity scaling (1.5ps) 10ps simulation with 1fs time steps

Supercell: c-axis Supercell: a-axis

Verification of Model Behavior Atoms well-behaved No-Diffusion MSD Velocity Distribution

Temperature Fluctuations Reasonable standard

deviation Energy Fluctuations Relative fluctuation

must be small

time [fs]

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Tem

pera

ture

[K]

220

240

260

280

300

320

340

360

time [fs]

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Ener

gy [e

V/a

tom

]-3.700

-3.698

-3.696

-3.694

-3.692

-3.690

-3.688

-3.686

-3.684

-3.682

Computational Approach

),(SeEE

Tk2dEdd 2

Binelastic

2

=

( , ( ))1( , )2

i t tS e e dt

= Use NJOY/LEAPR

=

3

() = (0) () =1(0) ()

1

Velocity Auto-Correlation Function

time [fs]

0 200 400 600 800 1000

VA

CF

[arb

.]

-1.0

-0.5

0.0

0.5

1.0

time [fs]

0 200 400 600 800 1000

VA

CF

[arb

.]

-1.0

-0.5

0.0

0.5

1.0

time [fs]

0 1000 2000 3000 4000 5000

VA

CF

[arb

.]

-1.0

-0.5

0.0

0.5

1.0

Velocity Auto-Correlation Function

time [fs]

0 2000 4000 6000 8000 10000

VA

CF

[arb

.]

-1.0

-0.5

0.0

0.5

1.0

Velocity Auto-Correlation Function

time [fs]

0 200 400 600 800 1000

VA

CF

[arb

.]

-1.0

-0.5

0.0

0.5

1.0

Velocity Auto-Correlation Function

time [fs]

0 200 400 600 800 1000

VA

CF

[arb

.]

-1.0

-0.5

0.0

0.5

1.0

time [fs]

0 1000 2000 3000 4000 5000

VA

CF

[arb

.]

-1.0

-0.5

0.0

0.5

1.0

Velocity Auto-Correlation Function

time [fs]

0 2000 4000 6000 8000 10000

VA

CF

[arb

.]

-1.0