Quasielastic Neutron Scattering

1. Quasielastic Neutron Scattering Miguel A. Gonzalez Institut Laue-Langevin (Grenoble, France) gonzalezm@ill.eu

2. Outline • General remarks and reminders • The main equations and their physical meaning • QENS models for translational diffusion and localized motions • The EISF and its physical interpretation • Instrumentation: A Neutron Backscattering spectrometer (IN16) • Examples • Complex systems and MD simulations • Conclusions and references

3. Neutron scattering: What can we see?

4. Coherent and incoherent neutron scattering • Incoherent scattering appears when there is a random variability in the scattering lengths of the atoms in the sample, e.g. different isotopes or isotopes with non-zero nuclear spin so (b+ = I + ½)  (b = I  ½) . • Coherent scattering: Information on spatial correlations (structure) and/or collective motion. – Elastic: Where are the atoms? What are the shape of objects? – Inelastic: What is the excitation spectrum in crystals – e.g. phonons? – Quasielastic: Correlated diffusive motions. • Incoherent scattering: Information on single-particle dynamics. – Elastic: Debye-Waller factor, Elastic Incoherent Structure Factor (EISF)  geometry of diffusive motion (continuous, jump, rotations) – Inelastic: Molecular vibrations – Quasielastic: Diffusive dynamics, diffusion coefficients. Here focus on quasielasticincoherent neutron scattering (QEINS or QENS) !

5. When will we have incoherent neutron scattering? • Mainly incoherent scatterers: • H • 49Ti • V • 53Cr • Co • Sm Or if polarized neutrons are used to separate coherent and incoherent scattering! From Jobic & Theodorou, Micropor. Mesopor. Mater. 102, 21-50 (2007)

6. EINS and QEINS: Main information Elastic intensity Quasielastic intensity Quasielastic broadening Debye-Waller factor: Vibrational amplitudes A0 = EISF (ratio elastic/total): Geometry of motion Width: Characteristic time scale From Heberle et al., Biophys. Chem. 85, 229-248 (2000)

7. A true QEINS spectrum: water Teixeira et al., Phys. Rev. A31, 1913 (1985) Qvist et al., J. Chem. Phys. 134, 144508 (2011) IN6@ILL IN5@ILL • Neutron exchanges small amount of energy with atoms in the sample: Typically from 0.1 eV (BS) to 5-10 meV (TOF). • • Vibrations normally appear just like flat background and treated as Debye-Waller. • • Maximum of intensity is at = 0. • • Low-Q – typically < 5 Å1 and often <2-3 Å1. PELICAN@ANSTO

8. Instrumental constraints • The instrumental resolution and the dynamical window (maximum • energy transfer) determine the observable timescales: • IN16:   1 eV min 0.1 eV tmax 2/min40 ns • max 15 eV tmin 275 ps • IN13:   8 eV min 1 eV tmax 4 ns • max 100 eV tmin 40 ps • IN5:   50 eV min 5 eV tmax 800 ps • max 10 meV tmin 0.4 ps • The Q-range determines the spatial properties that are observable. • Typical range (IN16, IN5) is  0.2 – 2 Å1 3 – 30 Å. • In IN13, Qmax  5 Å1 dmin 1 Å. • Instrumental limitations (limited Q-range, resolution and energy range) • together with the complexity of the motion(s) can make interpretation • difficult.

9. QEINS is associated with relaxation phenomena, such as translational diffusion, molecularreorientations, confinedmotion within a pore, hopping among sites, etc But how is related the QEINS signal or broadening with the physical information of interest to us?

10. Master equation We can measure the double differential cross section, i.e. the number of neutrons scattered into a detector having a solid angle  and with an energy between  and +d and this can be easily related to the dynamical structure factor, S(Q,), which is a correlation function related only to the properties of the scattering system. DIRECT RELATION: Measured quantity Physical information d2/dd S(Q, ) intermediate scattering function, I(Q,t)

11. FT in time FT in space Sinc(Q, ) Iself(Q,t) Gself(r,t) [energy]1[][volume]1 Self correlations (incoherent scattering) self intermediate function

12. Physical meaning of Gself(r,t) Gscl(r,t)dr is the probability that, given a particle at the origin at time t=0, the same particle is in the volume dr at the position r at time t ! From “Neutron and X-ray spectroscopy” (Hercules school)

13. Gs(r,0) = (r) Gs(r,t ) 1/V Properties of Gself(r,t), Iself(r,t) and Sinc(Q,)

14. FROM THE GENERAL EXPRESSION TO USEFUL MODELS

15. Self intermediate scattering function

16. Approximations or assumptions • A full analytical evaluation of Is(Q,t) is impossible* unless we assume that we can separate motions having different time scales and neglect any coupling between them: • Vibrations: internal (molecule), external (lattice vibrations). • Local motions: local diffusion, molecularreorientations. • Translational diffusion. • This is valid to separate vibrations from translations or rotations, as they have very different time scales (typically 1014 s for vibrations and 1012 -1011 s for diffusive motions, either reorientations or translational diffusion). • Separating translational and rotational diffusive motions is less satisfactory, but nevertheless accepted in most cases as the only way to proceed (again the importance of roto-translational coupling in the experimental spectra can only be judged from computer simulations, e.g. work of Liu, Faraone and Chen on water). • * Is(Q,t) can be computed without approximations from a computer simulation trajectory (as we have r(t) for all atoms). This can be compared to experimental results, but there is not yet a direct way to refine it using the experimental S(Q,).

17. Self intermediate scattering function and incoherent dynamical structure factor

18. Vibrational terms

19. A first (too general) expression to fit to our data Adding instrument resolution and assuming that vibrations appear as flat background: Sinc(Q,) = B(Q) + eu2Q2 [ST(Q,)  SR(Q,)]  R(Q,)

20. Brownian motion E.g. liquid argon: Very weak interactions + small random displacements. Collisions are instantaneous, straight motion between them and random direction after collision. If Q is low enough to loose the details of the jump mechanism (because we look to a large number of jumps) we can use the same expression used to describe macroscopic diffusion (Fick’s law).

21. Translational diffusion (Brownian motion) Fick’s 2nd law tells how diffusion causes concentration to change with time: We can arrive to an equivalent expression by introducing P(l,), which is the probability of a particle travelling a distance l during a time , after a collision: And we have the following conditions:

22. Translational diffusion (Brownian motion) FT in space FT in time Neutron spectrum is a lorentzian function with a width increasing strongly with Q: HWHM = DQ2.

23. Translational diffusion (Chudley-Elliott model) • Model for jump diffusion in liquids (1961). • Atoms or molecules ‘caged’ by other atoms and jumping into a neighbouring cage from time to time. • Jump length l identical for all sites. • Can be applied to atom diffusion in crystalline lattices. l = 1 Å D = 0.1 Å2meV = 1.519 × 105 cm2/s

24. Jump diffusion in cubic lattices • Lattice constant a and coordination number z = 6. • Jump vectors (a, 0, 0), (0, a, 0), and (0, 0, a). • If crystal oriented with x-axis parallel to Q:  = 1 meV1 = 0.658 ps

25. Localized motion • Hopping between 2 or more sites, e.g. CsOHH2O, crystals, … • Intramolecular reorientations, e.g. CH3 jumps, motion of side groups in polymers and proteins, … • Molecular rotations, e.g. plastic crystals, liquid crystals, … • Confined motion, e.g. in a pore All such motions are characterized by the existence of a non-null Q-dependent elastic contribution  elastic incoherent structure factor (EISF).

26. Jump model between two equivalent sites  r1 r2 And assuming that at t = 0, the atom is at r1: Solutions are:

27. Jump model between two equivalent sites r1 r2

28. Jump model between two equivalent sites r1 r2 d If powder, average over all possible orientations EISF QISF

29. Jump model between two equivalent sites r1 r2 d A0d(w) Half width ~1/ (independent of Q) w 0

30. HWHM EISF A0(Q)= ½[1+j0(Qd)/(Qd)] 1 ½ Q Qr Jump model between two equivalent sites r1 r2 d

31. EISFs corresponding to different rotation models

32. EISFs and widths of different rotation models

33. Physical meaning of the EISF And the EISF is easily obtained as the ratio between the elastic intensity and the total (elastic + quasielastic, no DW) intensity: And if the system is in equilibrium, there are no correlations between positions at t=0 and t=, so: Direct information about the region of space accessible to the scatterers(Bee, Physica B 182, 323 (1992))

34. Physical meaning of the EISF 2/Q If the atom moves out of the volume defined by 2/Q in a time shorter than tmax set by the instrument resolution it will give rise to some quasielastic broadening  loss of elastic intensity. The EISF is essentially the probability that a particle can be found in the same volume of space after the time tmax.

35. The EISF can be obtained without any ‘a priori’ assumption and compared to any of the many physical models available in the literature (see M. Bee: “Quasielastic Neutron Scattering”, 1988). In this way we can determine the geometry of the motion that we observe and then apply the correct model to obtain the characteristic times. Caveat: In complex systems this is not a trivial task and can be even impossible. In such cases it is useful to recourse to computer simulations.

36. A NEUTRON BACKSCATTERING SPECTROMETER: IN16

37. Backscattering is a special kind of TAS

38. Best resolution when 2 = 180 (backscattering)

39. BS instruments in the practice

40. IN16 at ILL Si(111) Si(111)

41. Performing an energy scan - Move monochromator with velocity vD parallel to reciprocal lattice vector . - Energy of reflected neutrons modified by a longitudinal Doppler effect (the neutrons see a different lattice constant in case of a moving lattice). - Register scattered neutrons as a function of Doppler velocity vD. - Maximum achievable speed determines max energy transfer (~10-40 eV) - Or change the lattice distance of the monochromator by heating/cooling. - Need crystals having a large thermal expansion coefficient, good energy resolution and giving enough intensity. - Possible energy transfers > 100 eV

42. IN16: Resolution better than 1 eV

43. Fixed window scan: Measure S(Q,~0) Obtain an effective mean square displacement! Dynamical transition in proteins (Doster et al., Nature 1989)

44. Low-frequency excitations Nuclear hyperfine splitting of Nd Tunnelling spectrum of NH4ClO4 and with different levels of partial deuteration Probe potential energy barriers and rotational potentials (test for simulations)

45. Quasielastic scattering: motions in a polymer

46. EXAMPLES

47. - Dislocation pipe diffusion  enhanced atomic migration along dislocations due to a reduced activation barrier. - Can improve diffusivity by orders of magnitude.

48. Hydrogen diffusion in Pd QENS spectra (BASIS, SNS) & fits Line widths (Chudley-Elliot model)  l &  - D is lower by 2-3 orders of magnitude compared to regular bulk diffusion. - Diffusivities for hydrogen DPD characterized by much lower Ea. Heuseret al., PRL 2014

49. Hydrogen diffusion in Pd - Suggest existence of a continuum of lattice sites associated with dislocations. - Reduced site blocking. - H de-population of dislocation trapping sites goes as ekT bulk regular diffusion above 300 K. - DFT shows metastable sites characterized by a lower activation energy for diffusion. - DPD expected to depend on H concentration and dislocation density. (QENS ~ 230 meV) (QENS ~ 40-80 meV) QENS represent a unique experimental scenario that allows the diffusivity associated to dislocation pipe diffusion to be directly quantified! Heuseret al., PRL 2014

50. - Fe(pyrazine) [Pt(CN)4]  spin crossover (SCO) compound. - Neutron diffraction points to free rotations of the ligand in the HS, which are blocked in the LS.