Diffusion of Hydrogen in Materials: Theory and Experiment

1. Diffusion of Hydrogen in Materials: Theory and Experiment • Outline • Diffusion—Fick’s 1st and 2nd laws; microscopic view. • Diffusion measurement techniques—some data on hydrogen diffusion. • Neutron scattering theory Part 1—coherent and incoherent. • Examples of scattering—Dispersion curves and vibrational density of states. • Neutron scattering theory Part 2—Incoherent quasi-elastic neutron scattering (QENS). • QENS—instruments, data related to the diffusion of hydrogen. Brent J. Heuser University of Illinois, Urbana, IL 2007 LANSCE Neutron School

2. buy this book!! Useful books related to neutron scattering in general and QENS in particular:

3. C(x) Diffusion coefficient J1 1st Law J2 x J(x) J1 J2 x 2nd Law J1 J2 Dx Diffusion—Fick’s 1st and 2nd Laws—Macroscopic Point of View Diffusion acts to even out concentration gradients and determines kinetic response of system:

4. Thin-film solution Diffusion-couple solution C’ Separation of variables solution—diffusion out of a slab C’ h Solutions to Diffusion Equation e-folding time: t = x2/4D

5. Impurity interstitials—like hydrogen Vacancy self-diffusion hydrogen thermal vacancy Classical picture—transition state theory yields jump frequency over saddle point saddle point Interstitial self-diffusion-- these arrangements are called crowdions Diffusion—Microscopic point of view Diffusion processes at microscopic scale coupled to lattice defects in crystalline solid

6. Classical Diffusion Arrhenius behavior; Q is classical barrier height Do Q VSD—vacancy self diffusion Diffusion of regular (non-impurity) atoms requires the presence of vacancies. Easier to form vacancies Easier for interstitials to move VSD vacancy interstitial n is the vibrational or Debye frequency The Diffusion Coefficient Physics and temperature dependence come into problem via D

7. Useful books related to hydrogen diffusion in metals

8. Effect of size Small polaron theory of H diffusion in bcc metals Anomalous isotope effect in fcc metals Examples of Diffusion Data Diffusion of hydrogen in metals: quantum effects such as tunneling and phonon-assisted processes are important.

9. Elastic (ħw=0) Inelastic (ħw≠0) Incoherent Coherent SANS Diffraction Reflectometry QENS Coherent Scattering Vibrational Spectroscopy Phonon Dispersion Focus on this one for diffusion. Hydrogen has very large incoherent cross section— therefore incoherent QENS. Neutron Scattering Coherent scattering—collective phenomena involving different nuclei that gives rise to interference effects. Incoherent scattering—scattering from the same nuclei at different times; no interference effects.

10. Double differential scattering cross section Coherent term—double summation over all nuclei that depends on correlation of the same nucleus at different times and different nuclei at different times. Incoherent term—single summation over all nuclei that depends on correlation of the same nucleus at different times.

11. Incoherent and coherent cross sections • Two contributions to incoherent scattering: • non-zero nuclear spin • presents of more than one isotope

12. General Picture QENS Phonon dispersion curves in PdD0.6 optical modes due to hydrogen Phonon dispersion curves in NbD0.6 and TaD0.22 acoustic modes Dispersion curves (coherent) and vibrational density of states (incoherent) Local harmonic oscillator potential Hydrogen vibrational DoS

13. SNS Minimum energy resolution Dw determines maximum coherence time tcoh of the measurement via: where (sinx)/x is a “transmission” function in time and tcoh~1/Dw. NIST Mapping of physical processes and instrument coverage in Q-w space.

14. Intermediate Scattering Functions This is measured in scattering exp. Structure Factors Connection to Differential Cross Section Basic Theory of Neutron Scattering

15. different particle Differential probability that given particle at origin at t=0, any particle will be at position r at time t. same particle van Hove self-correlation function van Hove Correlation Function

16. Translation Jump Diffusion in Bravais Lattice Chudley-Elliott model Considering self diffusion QENS measures L

17. Limiting Cases of L(Q)

18. Hydrogen diffusion via o-o jumps Polycrystalline Pd Hydrogen diffusion via o-o jumps Single crystal Pd Hydrogen diffusion in Nb—bcc metal with small polaron hopping Hydrogen diffusion in simple metals

19. Instruments for QENS—NIST HFBS

22. DCS BS

23. SNS Minimum energy resolution Dw determines maximum coherence time tcoh of the measurement via: where (sinx)/x is a “transmission” function in time and tcoh~1/Dw. NIST Mapping of physical processes and instrument coverage in Q-w space.