1. Diffusion of Hydrogen in Materials: Theory and Experiment Brent J. Heuser University of Illinois, Urbana, IL 2007 LANSCE Neutron School Outline Diffusion—Fick’s 1 st and 2 nd 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.

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

3. Diffusion—Fick’s 1 st and 2 nd Laws—Macroscopic Point of View Diffusion acts to even out concentration gradients and determines kinetic response of system: C(x) x J1J1 J(x) x J2J2 J1J1 2J22J2 J1J1 J2J2 xx 1 st Law 2 nd Law Diffusion coefficient

4. Solutions to Diffusion Equation Thin-film solution Diffusion-couple solution C’ Separation of variables solution—diffusion out of a slab h C’ e-folding time: t = x 2 /4D

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

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

7. Useful books related to hydrogen diffusion in metals

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

9. Neutron Scattering Elastic (ħ  =0) Diffraction SANS Reflectometry Coherent 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. Inelastic (ħ  ≠ 0) Coherent Incoherent QENS Phonon Dispersion Vibrational Spectroscopy Focus on this one for diffusion. Hydrogen has very large incoherent cross section— therefore incoherent QENS.

10. Double differential scattering cross section Incoherent term—single summation over all nuclei that depends on correlation of the same nucleus at different times. Coherent term—double summation over all nuclei that depends on correlation of the same nucleus at different times and different nuclei 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. Dispersion curves (coherent) and vibrational density of states (incoherent) Phonon dispersion curves in NbD 0.6 and TaD 0.22 Phonon dispersion curves in PdD 0.6 acoustic modes optical modes due to hydrogen Hydrogen vibrational DoS Local harmonic oscillator potential General Picture QENS

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

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

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

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

17. Limiting Cases of  (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. BS DCS

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