Progress of Thermal Scattering Law Development and ......Progress of Thermal Scattering Law...

Progress of Thermal Scattering Law Development and Evaluations

at North Carolina State University

Cole Manring, Colby Sorrell, Ben LarameeAyman I. Hawari

Nuclear Reactor Program

Department of Nuclear Engineering

North Carolina State University

Raleigh, North Carolina, USA

Technical Program Review

Nuclear Criticality Safety ProgramMarch 26 – 27, 2019 • Amarillo, TX, USA

Acknowledgement

The many graduate students, postdocs, andresearch staff at North Carolina StateUniversity

Collaboration with LLNL and Bettis labs◼ David Heinrichs, Michael Zerkle, Jesse Holmes

Funding◼ US NNSA Nuclear Criticality Safety program

◼ US Naval Nuclear Propulsion Program

FY 2018/2019 4 new TSL evaluations

◼ 3 first-of-a-kind evaluations

Modern predictive methods for thermal neutron crosssection calculations based on the use of atomisticsimulations

◼ Ab initio lattice dynamics

◼ Molecular dynamics (ab initio and classical) New materials

All states of matter (solid, liquid, gas)

Imperfect structure

Expanding the FLASSH thermal scattering analysis platform◼ Removed approximations such as incoherent approximation,

cubic approximation, atom site approximation and SCT approximation

Initiated the integration of the Generalized Nuclear Data Structure (GNDS) format into FLASSH

Objectives

Objectives

Completed light water (H2O) Finalizing molten salt FLiBe

( ) ( )2

coh incoh s

d σ 1 ES κ,ω S κ,ω

d dE 4π E

= +

)()()( ω,κSω,κSω,κS ds

+=

is composed of two parts)( ω,κS

The scattering law

Using first Born approximation combined with Fermi pseudopotential, it can

be shown that the double differential scattering cross section has the form

Van Hove’s space-time formulation

( ) ( ) ( )1

, ,2

i r tS G r t e drdt

−

− −

=

where G( Ԧ𝑟,t) is the dynamic pair correlation function and can be expressed in terms of time dependent atomic positions.

Neutron Thermalization

2

( , )2

s

Binelastic

d ES

d dE k T E

=

Tk

EE

B

−= Energy transfer

Tk

)cosEE2EE(

B

−+= Momentum transfer

−

−−= dteeS ttis)(

2

1),(

deet ti 2/ 1

)2/sinh(

)(

2)(

−

−−=

() – density of states (e.g., phonon frequency distribution)

The scattering law (TSL) is the Fourier transform of a Gaussian correlation

function

( ) ( )s B sS , k T S κ,ω =

Since 1960s

GASKET

NJOY/LEAPR

INCOHERENT

APPROXIMATION

Thermal Scattering Law Analysis

Key development in the last 20 years isthe use of atomistic simulations methodsto support the evaluation process

◼ Produce data necessary to calculate the TSL including

DOS for evaluation of TSL

Direct access to TSL using correlation analysis

ENDF/B-VIII TSL EvaluationsMaterial ENDF Library Name Evaluation

BasisInstitution

Beryllium metal tsl-Be-metal.endf DFT/LD NCSUBeryllium oxide (beryllium) tsl-BeinBeO.endf DFT/LD NCSUBeryllium oxide (oxygen) tsl-OinBeO.endf DFT/LD NCSULight water (hydrogen) tsl-HinH2O.endf MD CABLight water ice (hydrogen) tsl-HinIceIh.endf DFT/LD BAPLLight water ice (oxygen) tsl-OinIceIh.endf DFT/LD BAPLHeavy water (deuterium) tsl-DinD2O.endf MD CABHeavy water (oxygen) tsl-OinD2O.endf MD CAB

Polymethyl Methacrylate (Lucite)

tsl-HinC5O2H8.endfMD NCSU

Polyethylene tsl-HinCH2.endf MD NCSUCrystalline graphite tsl-graphite.endf MD NCSU

Reactor graphite(10% porosity)

tsl-reactor-graphite-10P.endf

MD NCSU

Reactor graphite(30% porosity)

tsl-reactor-graphite-30P.endf

MD NCSU

Silicon carbide (silicon) tsl-CinSiC.endf DFT/LD NCSUSilicon carbide (carbon) tsl-SiinSiC.endf DFT/LD NCSUSilicon dioxide (alpha phase) tsl-SiO2-alpha.endf DFT/LD NCSUSilicon dioxide (beta phase) tsl-SiO2-beta.endf DFT/LD NCSUYttrium hydride (hydrogen) tsl-HinYH2.endf DFT/LD BAPLYttrium hydride (yttrium) tsl-YinYH2.endf DFT/LD BAPLUranium dioxide (oxygen) tsl-OinUO2.endf DFT/LD NCSUUranium dioxide (uranium) tsl-UinUO2.endf DFT/LD NCSUUranium nitride (nitrogen) tsl-NinUN.endf DFT/LD NCSUUranium nitride (uranium) tsl-UinUN.endf DFT/LD NCSU

Thermal Scattering Cross-Sections

Evaluation DFT/LD

PHONONVASP

Optimize the

system structure

Generate

displacements

Evaluate

Hellmann–

Feynman Forces

Evaluate the

Dynamical Matrix

Sample the

Brillouin Zone

DFT

Pseudo-

potential

Quantum

Mechanical

model

LD

Phonon

model

Harmonic

potential

Density of StatesThermal Scattering

Law S(α, β)

FLASSHThermal Scattering Cross Section

Thermal Scattering Cross-Sections

Evaluation MD/QM

Equilibrate the

systemGenerate particle

trajectories

Velocity Auto-

Correlation

Function (VACF)

Evaluate classical

Gcl(r, t) and Icl(k, t)

LAMMPS MD

Pair potential

Classical system

Density of States

Thermal Scattering

Law S(α, β)

Quantum Correction

Thermal Scattering

Cross SectionFLASSH

Computational Capabilities Hybrid mini cluster - 17 nodes

▪ 324 CPU cores▪ 22 Nvidia GPUs▪ Expanding…..

Parallel computations▪ Atomistic simulations▪ TSL analysis▪ Neutronic simulations▪ System design

VASP, PHONON, LAMMPS PREPRO, NJOY, FUDGE, SAMMY, MCNP, Serpent, GEANT4, McStas, PARET, RELAP, COMSOL

Heavy Paraffinic Oil

0.0 0.1 0.2 0.3 0.4 0.5 0.60

5

10

15

20

25

30

DO

S (

eV-1

)

Energy (eV)

0.000 0.001 0.0020.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

DO

S (

1/e

V)

Energy (eV)

0 500 1000 1500 2000-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0

VA

CF

(ar

b)

Time (fs)

Light Water

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

1

2

3

4

5

6

7

8

9

10

DO

S (

eV

-1)

Energy (eV)

283K 93 bond

293K 93 bond

300K 93 bond

323K 94 bond

350K 94 bond

373K 94 bond

400K 94 bond

423K 94 bond

450K 94 bond

473K 94 bond

500K 94 bond

523K 94 bond

550K 94 bond

573K 94 bond

600K 94 bond

623K 94 bond

650K 94 bond

12 6

2inter intra 214 1 exp ( ) ( )

2i j i j

i jtot

r OH eq eq

i j i jO O O O ij

q qU U U D r r K

r r r

= + = − + + − − − + −

Flexible TIP4P/2005 potential

Light Water

0.02 0.04 0.06 0.08 0.100.96

0.98

1.00

1.02

1.04

1.06

Ratio

Energy (eV)

NCSU 473K/293K

ENDF8 473K/293K

ENDF7 450K/293K

Dritsa [EXFOR #22613] (473 K / 293 K)

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

10-40

10-38

10-36

10-34

10-32

10-30

10-28

10-26

10-24

10-22

10-20

10-18

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

102

104

106

S(a

lpha,b

eta

)

Beta

8.19721E-8

1.31155E-6

3.27888E-5

1.31155E-4

0.00118

0.01312

0.11804

1.06236

10.28258

100.4158

819.72081

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 10410-4010-3810-3610-3410-3210-3010-2810-2610-2410-2210-2010-1810-1610-1410-1210-1010-810-610-410-2100102104106108

1010

S(a

lpha

,be

ta)

Alpha

0

1E-6

1.02345E-5

1.04745E-4

0.001

0.01027

0.10507

1.0062

10.29797

105.39471

398.10717

1E-5 1E-4 0.001 0.01 0.1 1 1010

100

1000

Tota

l X

S (

ba

rns)

Energy (eV)

283K

293K

300K

323K

350K

373K

400K

423K

450K

473K

500K

523K

550K

573K

600K, 80 atm

623K, 100 atm

650K, 150 atm

File 7

Liquid FLiBe – FY 2019 Eutectic with a mixture of

2:1 ratio of LiF and BeF2

Melting Point: 732K Boiling Point:1703K

DFT and MD analysis (with QM corrections)

TSL evaluation between 750K and 1500K

0.00 0.05 0.10 0.15

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

De

nsity o

f S

tate

s (

1/e

V)

Energy (eV)

Li

Born-Mayer potential

Hydrofluoric Acid

Strong hydrogen bonding

Unique molecular structure dynamics

Example CMD Potential:

LJ + 3 pt. chg.Perturbed Morse Osc.

𝑉 𝑟 = 𝐶 𝑒𝑥𝑝 −𝐷 𝑟 − 𝑟0 − 12 + 𝐸𝑟𝑉 𝑟 = 𝐴

𝐵

𝑟

12

−𝐵

𝑟

6

+⋯

FLASSH Code

FLASSH Code FeaturesNJOY FLASSH

Coherent Inelastic

Remove Incoherent Approximation

Remove Short Collision Time (SCT) Approximation

Integral against alpha differential cross section

NumericalDefault: Analytical

Optional: Numerical

a, B Gridding User inputDefault: Automatic grid

Optional: User input

Parallel Computing Using OpenMP

Graphite User Interface

Syntax and Error Checking

Coherent Elastic Calculation Supported Structure Hexagonal, FCC, BCC Any crystal structure

Supported MaterialsGraphite, Beryllium, Beryllium Oxide, Aluminum, Lead, Iron

Any material

Compound Materials 2 elements with ratio 1:1Any number of elements

with any ratio

Remove Cubic Approximation

Remove Atom Site Approximation

Coherent Elastic Scattering Cross Section

Over Ewald SphereOn every reciprocal lattice

point

Need to modify source code if calculating other materials

Yes No

Liquid Physics in FLASSH

Separation of the diffusive DOS from the continuous (solid) DOS in LEAPR

Convolution of the solid and liquid TSL components

𝑆𝑡𝑜𝑡𝑎𝑙 𝛼, 𝛽 = 𝑆𝑑𝑖𝑓𝑓. 𝛼, 𝛽 ∗ 𝑆𝑐𝑜𝑛𝑡. 𝛼, 𝛽 𝛽

Liquid Physics in FLASSHConstruct fine beta grid (for convolution):

Call convolve_grid subroutine to determine appropriate resolution and lower/upper beta limits

Build liquid TSL model over the new beta grid:

Call liquid model function (e.g., lang for Langevin)

◼ Call besk1 (Bessel) function if necessary

Interpolate solid TSL onto new beta grid:

Call interp_grid subroutine to interpolate values for every convolution ‘window’

Convolve the liquid and solid components:

Call convolve subroutine to perform the convolution

Construct total TSL:

Add in extra DW term

Output results:

Write TSL components to various files

Free Gas◦ Assumes a Maxwellian velocity distribution

Crystal Lattice◦ Compound nucleus effects separated from lattice effects

◦ Transition probability

◦ Self Scattering Law

Identical to that used in thermal scattering

Describes the energy-momentum phase space of a material

Doppler Broadening

23

( )( )

2

20

'1 '' ( ', ) ( ') , ( ', ) expFG FG FG

E EEE dE S E E E S E E

E

− −= =

( )( ) ( )

2

0

2 2

0

( , )

4 / 2

s

B

SE d

E E k T

−

=

− − +

Ab initio lattice dynamics◦ Predictive density of states (DOS)

◦ Current DOS implemented in the ENDF/B-VIII.0 cross section library for U in UO2

Doppler Broadening

24

◦ Fluorite Structure

◦ 2x2x2 supercell

◦ GGA-PBE+U

Ab initio lattice dynamics◦ Predictive density of states (DOS)

◦ Current DOS implemented in the ENDF/B-VIII.0 cross section library for U in UO2

Doppler Broadening

25

◦ Fluorite Structure

◦ 2x2x2 supercell

◦ GGA-PBE+U

FLASSH Generalized TSL

Full Equation Cubic Approximation( )2 21

3sse k =

2

0

(2 th

2 2

)co

m s

B

W dM k T

k

=

Function of the density of states ρ(ω) Function of the polarization vector and dispersion relations

2

2 coth2 2

s

s

s

s B

eW

MN k T

=

( )2 0 20

2( e)

!

1,

2

U U

s

i

n

t

n

V

n

We e dtW

S Gn

+

−−

=

= =

( )( )

2

/2

1

1 1

(0) 2 sinh / 2

s

sB

e eG e

N k T

−= ( )( )

/ 2

1

1 ( )

2 sinh / 2G e

−=

5.49E-3 0 0

0 5.49E-3 0

0 0 5.49E-3


27

Beryllium Metal

o HCP (P63/mmc)

o Ab initio lattice dynamics

o 4x4x3 Supercell Exact Debye-Waller Matrix (Å2)

5.76E-3 -1.58E-5 1.01E-6

-1.58E-5 5.74E-3 -5.18E-7

1.01E-6 -5.18E-7 5.02E-3


28

Beryllium Metal

o HCP (P63/mmc)

o Ab initio lattice dynamics

o 4x4x3 Supercell Exact Debye-Waller Matrix (Å2)

FUDGE/GNDS Ongoing

collaboration with LLNL

Summary FY 2018

◼ 10 new TSL evaluations contributed to ENDF/B-VIII.0


FY 2019◼ 4 new TSL evaluations


Modern predictive methods for thermal neutron cross sectioncalculations based on the use of atomistic simulations

◼ Ab initio lattice dynamics

◼ Molecular dynamics (ab initio and classical) New materials

All states of matter (solid, liquid, gas)

Imperfect structure

FLASSH is a new thermal scattering analysis platform that uses a generalized theoretical approach for TSL calculations◼ Removed approximations such as incoherent approximation, cubic

approximation, atom site approximation and SCT approximation

Progress on FY 2019 tasks

Progress of Thermal Scattering Law Development and ......Progress of Thermal Scattering Law...

Documents

Transcript of Progress of Thermal Scattering Law Development and ......Progress of Thermal Scattering Law...