Thermal Neutron Scattering in Graphite

Transcription

1 Thermal Neutron Scattering in Graphite by Iyad I. Al-Qasir Department of Physics University of Jordan Amman, Jordan Supervisor Dr. Ayman I. Hawari Department of Nuclear Engineering North Carolina State University Raleigh, NC

2 Introduction Generation IV Very High Temperature Reactor (VHTR) are graphite moderated and gas cooled thermal spectrum rectors The characteristics of the low energy (E < 1eV) neutron spectrum in these reactors will be dictated by the process of neutron slowing-down and thermalization in the graphite moderator The ability to accurately predict this process in these reactors can have significant neutronic and safety implications Currently used libraries (ENDF/B-VII) are a product of the 1960s and remain based on many physical approximations, these libraries show noticeable discrepancies with experimental data June 13, 008

3 Graphite Inelastic Scattering Cross Section??? June 13, 008 3

4 Objectives To Review the currently used thermal neutron scattering laws of graphite as a function of temperature To Update models and models parameters by introducing - The new developments in solid-state physics (ab initio approach) - The coherent part of the inelastic scattering To Develop and generate new sets of temperature dependent thermal neutron scattering laws June 13, 008 4

5 Outline Neutron Thermalization Lattice Dynamics Results Conclusions June 13, 008 5

6 Neutron Thermalization June 13, 008 6

7 Graphite Cross Section Thermal Neutrons 10 1 Cross Section (b) The wavelength of thermal neutrons is comparable to the interatomic distances in solids and liquids The energy of thermal neutrons is of the same order of the excitations in condensed matter (e.g. phonons in crystalline materials) Energy(eV) June 13, 008 7

8 Inelastic double differential scattering cross section d σ 1 k = σ dω de 4π k S ( ) ( ) P κ, ω + ( σ + σ ) S κ, ω P coh d coh incoh P= 1 P= 1 s Graphite σ coh = 5.50b σ incoh ~ 0b Nuclear Part Scattering Law S( α, β ) = k B Te β / S( κ,ω) Momentum transfer Energy Transfer E + E μ α = Ak T β = E - E k T B B EE Atomic (Dynamical) Part June 13, 008 8

9 Approximations 1- Incoherent approximation: S d ( α,β ) = 0 d σ de dω σ coh + σ 4π k T B incoh E E e -β P= 1 P S s ( α, β ) - Harmonic interatomic forces 3- Monoatomic solid 4- The solid has one atom per unit cell 5- The unit cell has a cubic symmetry 6- The solid vibrational modes are described by a continuous spectrum, called the phonon frequency distribution ρ(β) 7- Gaussain Approximation S s ( α, β ) = 1 π γ ( τ ) June 13, e iβ τ e dτ

10 γ ( τ ) = α βτ ρ( β )[1 e i ] e β sinh( β ) β dβ In order to calculate the scattering law, all we need is the phonon frequency distribution ρ(β) (Lattice Dynamics) The above formulation represents the basis of computer programs such as GASKET and LEAPR/NJOY Discrepancies between measurements and calculations (differential and integral) June 13,

11 Lattice Dynamics June 13,

12 Output: Disersion elations: Polarization vectors: Phonon Distribution: ω ( q ) j e dj (q) ρ ( ω ) Phonon Frequency ρ ρ (β (β)) Lattice dynamics Force Force Constants Models Models Walter Khon:1998 Nobel Laureate in Chemistry for his development of the Density functional Theory Early Early Work Work -Force -Force constant values values are are obtained by by fitting fitting to to thermodynamical experimental data data (( e.g., e.g., specific specific heat, heat, comperessibility, ) comperessibility, ) Graphite (1965) (1965) Young-Koppel (ENDF/B-VII) Later Later (( Neutron Scattering )) force force constant obtained by by fitting fitting to to experimental dispersion curves curves along along symmetry directions in in the the 1 st st Brillouin Zone Zone Graphite (1970) (1970) Nicklowet et al, al, (ORNL) Recently (Ab (AbInitio Calculations) Based Based on on the the Density Density Functional Functional Theory Theory (DFT) (DFT) and and the the Pseudopotentiapotential Approximation Pseudo- Approximation Graphite (007) (007) Al-Qasir-Hawari (NSCU) DFT: Replace the many-electron problem by an exactly equivalent set of self-consistent one-electron equations June 13, 008 1

13 Results June 13,

14 Graphite Hexagonal structure Four atoms per unit cell a=b=.447å c= Å 6x6x1 supercell used June 13,

15 Graphite Dispersion Relations Ab initio vs. Experimental Data A Γ M K Γ 50 LO LO 0.0 TO TO Frequency (THz) LO' LO ZO L A ZO' ZA TA Ab initio 144atom Neutron Scattering X-ray Scattering HREELS He-atom scattering HREELS HREELS Raman scattering TA ZA L A ZO' ZO Energy (ev) Wave Vector (A -1 ) June 13,

16 Graphite Heat Capacity Heat Capacity (Jol/mol deg) NCSU DeSorbo, et al., (1953) Heat Capacity (Jol/mol deg) T(K) T(K) June 13,

17 NCSU vs. Young-Koppel (YK) Phonon Frequency distribution (1/eV) ZO' NCSU Young-Koppel (1965) ZA ZO+TA ZO TO LO Energy (ev) June 13,

18 Inelastic Scattering Cross Section June 13,

19 Scattering Law 10 1 NCSU ENDF/B-VII (YK) NWS 10 0 Carvalho (1968) β=0. T=533K S(α,β)/α Alpha June 13,

20 Cross Section (II) Coherent One-Phonon June 13, 008 0

21 June 13, Coherent One-Phonon Contribution ( ) ( ) + + = Ω = = 1 1, ) (, 4 1 P s P incoh coh P d P coh S S k k de d d ω κ σ σ ω κ σ π σ ( ) ( ) ( ) + + Ω = exact d s Incoh Approx P s P coh S S S k k de d d ω κ ω κ ω κ π σ σ,, ), ( Incoherent Approximation Include Coherent 1ph contribution 1 S= 1 S s + 1 S d ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } , 3 1 τ κ δ ω ω δ τ κ δ ω ω δ κ ω π ω κ τ κ = q n q n e e e M v S s s s s s d ds W d i s d N Coherent One Phonon Scattering Law x

22 Inelastic Scattering Cross Section June 13, 008

23 Scattering Law 10 1 NCSU ENDF/B-VII (YK) NWS 10 0 Carvalho (1968) β=0. T=533K S(α,β)/α Alpha June 13, 008 3

24 Scattering Law 10 1 NCSU-1P NCSU ENDF/B-VII (YK) 10 0 NWS Carvalho (1968) β=0. T=533K S(α,β)/α Alpha June 13, 008 4

25 Conclusions The ab initio direct approach was used to generate the dispersion relations, and phonon frequency distribution. Excellent agreements were observed between the dispersion relations, and heat capacity with experimental data Some improvement was observed in graphite cross section using the ab initio phonon frequency distributions Because graphite is a strong coherent scatterer, the scattering model was reformulated to account for the one phonon coherent scattering. Excellent agreement was found June 13, 008 5

26 Reviewing and Generation of thermal neutron scattering cross sections for different crystalline moderators such as Be, BeO, ZrH, ZrH 1.75 and ZrH 1.6 Generation of thermal neutron scattering cross sections for new materials such as ThH, CaH, Bi, Al O 3 Investigation of radiation effect on thermal neutron scattering in graphite Impact of Simple Carbon Interstitial formations on Thermal Neutron Scattering in Graphite Hawari, A. I., A. I., Al-Qasir, I. I, and Ougouag, A. M, Nucl. Sci. Eng. 155, (007) June 13, 008 6

27 Welcome to Jordan June 13,