Quasielastic Neutron Scattering Miguel A. Gonzalez Institut Laue-Langevin (Grenoble, France) [email protected] 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 Neutron scattering: What can we see? 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 quasielastic incoherent neutron scattering (QEINS or QENS) ! 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) EINS and QEINS: Main information Elastic intensity Debye-Waller factor: Vibrational amplitudes Quasielastic intensity A0 = EISF (ratio elastic/total): Geometry of motion Quasielastic broadening Width: Characteristic time scale From Heberle et al., Biophys. Chem. 85, 229-248 (2000) A true QEINS spectrum: water Teixeira et al., Phys. Rev. A 31, 1913 (1985) Qvist et al., J. Chem. Phys. 134, 144508 (2011) [email protected] [email protected] • 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 Å1 and often <2-3 Å1. [email protected] 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 Å1 3 – 30 Å. In IN13, Qmax 5 Å1 dmin 1 Å. • Instrumental limitations (limited Q-range, resolution and energy range) together with the complexity of the motion(s) can make interpretation difficult. QEINS is associated with relaxation phenomena, such as translational diffusion, molecular reorientations, confined motion within a pore, hopping among sites, etc But how is related the QEINS signal or broadening with the physical information of interest to us? 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 d2/dd Physical information S(Q, ) intermediate scattering function, I(Q,t) Self correlations (incoherent scattering) self intermediate function FT in time Sinc(Q, ) [energy]1 FT in space Iself(Q,t)  Gself(r,t) [volume]1 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) Properties of Gself(r,t), Iself(r,t) and Sinc(Q,) G self ( r , 0 ) ( r ) G self ( r , t )d r 1 I self ( Q , 0 ) 1 0 1 Gs(r,0) = (r) Gs(r,t ) 1/V S inc ( Q , )d 1 S inc ( Q , )d Q 2 2M FROM THE GENERAL EXPRESSION TO USEFUL MODELS Self intermediate scattering function 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, molecular reorientations. • Translational diffusion. • This is valid to separate vibrations from translations or rotations, as they have very different time scales (typically 1014 s for vibrations and 1012 -1011 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,). Self intermediate scattering function and incoherent dynamical structure factor Vibrational terms A first (too general) expression to fit to our data Adding instrument resolution and assuming that vibrations appear as flat background: 2Q2 Sinc(Q,) = B(Q) + eu [ST(Q,) SR(Q,)] R(Q,) 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). Translational diffusion (Brownian motion) Fick’s 2nd law tells how diffusion causes concentration to change with time: t G s (r , t ) D G s (r , t ) 2 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: t l 2 G s (r , t ) 6 D G s (r , t ) 2 And we have the following conditions: G s ( r ,0 ) ( r ) G s ( r , t ) dr 1 l 2 6 Translational diffusion (Brownian motion) G s ( r , t ) 4 Dt 3 2 exp r 2 / 4 Dt FT in space I s ( Q , t ) exp DQ t 2 FT in time S inc ( Q , ) 1 DQ 2 2 DQ 2 2 Neutron spectrum is a lorentzian function with a width increasing strongly with Q: HWHM = DQ2. 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. S inc ( Q , ) (Q ) (Q ) 1 (Q ) 1 sin Ql 1 Ql 2 6D 2 l l =1Å D = 0.1 Å2meV = 1.519 × 105 cm2/s 2 sin Ql 1 Ql Ql 3 Ql 5 Ql 6D 3 ! 5 ! Ql 1 ( Q ) 2 1 DQ l Ql 2 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). (Q ) 1 cos Q x a • If crystal oriented with x-axis parallel to Q: = 1 meV1 = 0.658 ps 3 Localized motion • Hopping between 2 or more sites, e.g. CsOHH2O, 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). Jump model between two equivalent sites r1 t t t p ( r1 , t ) p ( r2 , t ) p ( r1 , t ) 1 r2 1 p ( r1 , t ) p ( r1 , t ) p ( r2 , t ) 0 1 1 p ( r2 , t ) p ( r2 , t ) p ( r1 , t ) p ( r2 , t ) 1 And assuming that at t = 0, the atom is at r1: Solutions are: p ( r1 , t ) A Be 2 t / p ( r1 , 0 ) A B 1 p ( r2 , t ) A Be 2 t / p ( r2 , 0 ) A B 0 Jump model between two equivalent sites r1 p ( r1 , t ; r1 , 0) p ( r2 , t ; r2 , 0) 1 2 1 [1 e [1 e r2 2 t 2 t ] ] 2 p ( r2 , t ; r1 , 0) p ( r1 , t ; r2 , 0) I ( Q , t ) [ p ( r1 , t ; r1 , 0) p ( r2 , t ; r1 , 0) e [ p ( r1 , t ; r2 , 0) e I (Q , t ) 1 2 iQ ( r1 r2 ) iQ ( r2 r1 ) 1 2 1 [1 e [1 e 2 t 2 t ] ] 2 ] p ( r1 , 0) p ( r2 , t ; r2 , 0)] p ( r2 , 0) [1 cos Q .( r2 r1 )] 1 2 [1 cos Q .( r2 r1 )] e 2 t Jump model between two equivalent sites r1 S (Q , ) 1 2 d r2 [1 cos Q .( r2 r1 )] ( ) 1 2 [1 cos Q .( r2 r1 )] 2 1 4 2 2 If powder, average over all possible orientations 1 sin Qd 1 sin Qd 1 2 S ( Q , ) 1 1 2 2 Qd 2 Qd 2 1 S ( Q , ) A0 ( Q ) A1 ( Q ) EISF QISF 1 2 1 2 2 1 2 1 2 Jump model between two equivalent sites r1 d r2 A0() Half width ~1/ (independent of Q) 0 Jump model between two equivalent sites r1 EISF d r2 HWHM A0(Q) = ½[1+j0(Qd)/(Qd)] 1 ½ Qr Q EISFs corresponding to different rotation models EISFs and widths of different rotation models Physical meaning of the EISF S ( Q , ) I ( Q , ) ( ) S qel (Q , ) EISF ( Q ) I ( Q , ) And the EISF is easily obtained as the ratio between the elastic intensity and the total (elastic + quasielastic, no DW) intensity: S (Q , ) el EISF ( Q ) I (Q , ) e iQ r ( ) e iQ r ( 0 ) I (Q , ) e iQ r ( ) S (Q , ) S el qel (Q , ) And if the system is in equilibrium, there are no correlations between positions at t=0 and t=, so: e iQ r ( 0 ) e iQ r ( ) 2 e iQ r ( 0 ) 2 Direct information about the region of space accessible to the scatterers (Bee, Physica B 182, 323 (1992)) 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. 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. A NEUTRON BACKSCATTERING SPECTROMETER: IN16 Backscattering is a special kind of TAS Best resolution when 2 = 180 (backscattering) BS instruments in the practice IN16 at ILL Si(111) Si(111) 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 IN16: Resolution better than 1 eV Fixed window scan: Measure S(Q,~0) Obtain an effective mean square displacement! Dynamical transition in proteins (Doster et al., Nature 1989) 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) Quasielastic scattering: motions in a polymer EXAMPLES - Dislocation pipe diffusion enhanced atomic migration along dislocations due to a reduced activation barrier. - Can improve diffusivity by orders of magnitude. 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. Heuser et al., PRL 2014 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 ekT bulk regular diffusion above 300 K. (QENS ~ 230 meV) - DFT shows metastable sites characterized by a lower activation energy for diffusion. - DPD expected to depend on H concentration and dislocation density. (QENS ~ 40-80 meV) QENS represent a unique experimental scenario that allows the diffusivity associated to dislocation pipe diffusion to be directly quantified! Heuser et al., PRL 2014 - 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. 295 K (IN5) HS LS Bz - Switching of rotation associated with change of spin state. - In HS, pz rings perform 4-fold jump motion about the coordinating N axis. - Correlation between rotation of pz and change of spin state practical element for creating artificial molecular machines. Rodriguez-Velamazan et al., JACS 2012 Bz (PyH)I EISF (6 equivalent sites) - Benzene and (PyH)I (at high-T) show a 6-fold potential with equivalent minima. - At low-T, (PyH)I has a different crystalline phase and NMR indicates that reorientations in this phase take place in an asymmetric potential. - MD in good agreement with QENS/NMR data and indicates that asymmetric potential is due to the formation of weak H-bonds N-HI. (PyH)NO3 @ IN10 (ILL) QENS MD MD snapshot for (PyH)NO3 @ 290 K - In (PyH)NO3 only two orientations are significantly populated. - Two-well asymmetric potential related to the two orientations where N-HO hydrogen bonds can be formed. - Picture confirmed by MD simulations. Pajzderska et al., JCP 2013 Dealing with complex systems … - In most cases, needmicroscopic some kind of computational model to understand How does the structure and and interpret the QENS spectra. dynamics change with varying alkyl chain length? - The most useful tool is MD (either using empirical potentials or using ab initio DFT to compute interatomic forces) Solve Newton’s equation for a molecular system: m(d2ri/dt2) = fi = u(r) - From the MD simulation we will get the trajectory of all the atoms in our model (typically 102 for DFT, 105 for classical MD) during the simulation time (typically several ps for DFT, hundreds of ns for classical MD) Compute all kind of properties and, in particular, I(Q,t). - Today there are many available tools that can help us doing this. Molecular dynamics in metallic and highly plastic compounds of polyaniline A study using quasi-elastic neutron scattering measurements and molecular dynamics simulations (Maciek Sniechowski, David Djurado, Marc Bee, Miguel Gonzalez …) H N Polyaniline – Emeraldine base (insulating form) N N H N reduced oxidized H N H+ N H H + N H N + Emeraldine salt (conducting form) Structural model Structural Analysis 3.5 Å ~25-38 Å Quasi-elastic neutron scattering (QENS) studies of polyaniline/DB3EPSA TOF Spectrometers: IN6 ILL (50-100eV) MIBEMOL LLB Saclay (85eV) IRIS at ISIS GB (15eV) - Classical QENS data analysis in terms of EISF - MD Simulations (Compass in Cerius/MS) -> S(Q,w), I(Q,t): comparison with experiment confirmation of theoretical model DB3EPSA Model of local diffusion of protons in spheres Volino and Dianoux, Mol. Phys.41, 271,(1980) R10 R9 j (Q .R m ) EISF 1 Q .R m R7 R8 R6 2 R m R a , ms R5 R4 R3 R1 The radii of spheres are distributed in size along the alkoxy tails according to a gamma function: 3 parameters to adjust for varying the curve shape! Dynamic structure factor Sinc(q,w) IN6 spectrometer S inc ( Q , ) R(q,t) : resolution function of IN6 1 I ( Q , t ) R ( Q , t ) exp( i t ) dt 2 0,081,0 q=2 A 0,8 T=235K simulation experiment T=280K simulation experiment T=310K simulation experiment T=340K simulation experiment S(q,) 0,060,6 S(q,) -1 0,4 0,2 0,04 0,0 -0,4 -0,2 0,0 0,2 0,4 -0,4 -0,2 resolution function E (meV) 0,02 0,00 -1,0 -0,8 -0,6 0,0 E (meV) 0,2 0,4 0,6 0,8 1,0 Analysis of the individual atom trajectories Mean square displacement: MSD ( t ) 2 bi ri ( t t 0 ) ri ( t 0 ) 2 i R10 DB3EPSA R10 R6 T=340 K R2 R6 R1 R0 PANI chain R2 R0 R1 MD confirms the model employed to fit the QENS spectra! A complex example containing several contributions … Room temperature ionic liquids based on the imidazolium How does the microscopic structure andcation dynamics change with varying alkyl chain length? EmimBr or C2mimBr BmimBr or C4mimBr HmimBr or C6mimBr QENS analysis F(Q,t) DW x T(Q,t) R(Q,t) L(Q,t) S(Q,) exp(Q2u2) × [T(Q, ) R(Q, ) L(Q, )] T(Q, ) L(T (Q)DQ2) R(Q, ) A0R + (1-A0R) L(R) L(Q, ) A0L + (1-A0L) L(L) S(Q,) A0RA0L L(T) + (1A0R) A0L L(T +R) + A0R (1A0L) L(T +L) + (1A0R) (1A0L) L(T +R +L) If R 0 (MD, NMR Imanari 2010) then: S(Q,) A0L L (T) + (1A0L) L (T +L) QENS: C2mimBr translational dynamics Data fitted with two lorentzians: 1 translational-like + 1 local-like 0.12 D D0 = (1.7±0.8).107m2s-1 Ea = 19 ± 2 kJmol-1 412K 0.09 Wt (meV) D follows Arrhenius law: = D0 exp(Ea / RT) with 393K 374K 0.06 354K 0.03 0.00 0 Reasonable agreement with NMR (Every, PCCP 2004), although D values 3-4 times larger. 1 2 2 2 Q (Å ) 3 T (K) D (1010 m2s1) 0 (ps) 353 2.7 ± 0.2 3.9 ± 0.6 373 3.4 ± 0.5 2.6 ± 0.4 392 5.1 ± 0.7 3.2 ± 0.2 412 6.6 ± 0.9 2.5 ± 0.2 4 QENS: C2mimBr local dynamics 250K 300K 354K 374K 393K 412K 1.0 A0(Q) 0.8 0.6 0.4 0.2 0.5 S(Q, E ) exp Q S (Q, E ) exp Q 2 2 u 2 u 2 1.0 -1 Q(A ) 1.5 A (Q ) ( E ) 1 A (Q )L W , E A (Q)L W , E 1 A (Q)L W , E 0 0 0 t r 0 t r 2.0 (solid) (liquid) QENS analysis using MD input (C2mimBr) EmimBr 3 0.5 2 R1 2 R (Å) 1 1 R2 0.75 4 (c) P 0 A0(Q) 3 0.4 0.2 160 240 320 400 480 T(K) 250K 300K 354K 374K 393K 412K 0.50 R2 R2 (d) 0.25 0.4 R1 0.8 1.2 1.6 -1 Q(Å ) (b) (a) 0.5 A (b) Crystal State (c) Liquid State 0.3 Probability 1.00 1 Aoun et al. , J. Phys. Chem. Letters 1, 2503 (2010) 2.0 2.4 2.8 Quasielastic widths: Simulation vs experiment Liquid 360K Liquid 360K Simulation: D (from width of narrow line) = 4.9 x 1010 m2/s vs 3.2 x 1010 m2/s obtained directly from m.s.d.! Crystal 300K EISF: Simulation vs experiment Qualitative or even semiquantitative agreement between experimental (fitted S(Q,)) and simulated (fitted F(Q,t) with equivalent model) widths and EISF’s. Simulated spectra and components COM trajectory Ring rotation (with fixed methyl + alkyl) Local motions (no COM or global rotation) Center of mass trajectory Self-diffusion coefficient consistent with value of D extracted directly from the mean square displacements. Global rotation EISF by groups When looking to individual groups, reasonable agreement with model of diffusion on the surface of a sphere. Local motions: Dihedral torsion Simulated spectra: EISF for chain motions Possible to fit to model of rotation in a circle. But meaningful? Local motions: Spatial distribution (in the crystal) C6 & C7 Methyl MD can give a much clearer picture of how the molecules really move, but they can also be misleading, so they should be validated using experimental data! CH2 in ethyl chain CH3 in ethyl chain CONCLUSIONS Or why should I use Quasi-elastic Neutron Scattering? • Applicable to wide range of scientific areas: – Biology: dynamic transition in proteins, hydration water, ... – Chemistry: complex fluids, ionic liquids, porous media, surface interactions, water at interfaces, clays, ... – Materials science: hydrogen storage, fuel cells, polymers, ... • Probes true “diffusive” motions. • Range of analytic function models systematic comparisons. • Close ties to theory – particularly Molecular Dynamics simulations. • Complementary to techniques such as light spectroscopy, NMR, dielectric relaxation, etc. • Unique – Can answer questions you cannot address otherwise: – (Q, ) information: provides information about the dynamics on length scales given by Q. – Very sensitive to H – Able to test microscopic models of motion and MD simulations – Large range of time scales: From sub-picosecond to several ns REFERENCES • Quasielastic Neutron Scattering, M. Bee (Bristol, Adam Hilger 1988) • Quasielastic Neutron Scattering and Solid State Diffusion, R. Hempelmann (Oxford University Press 2000). • Neutron and X-ray Spectroscopy, F. Hippert et al. (eds) (Springer 2006): Focused more on instrumentation. • Collection of articles from JDN8 school (Diffusion Quasiélastique des Neutrons): In french, but free access from SFN web page (www.neutronsciences.org Écoles thématiques) . • Quasielastic Neutron Scattering, G. R. Kneller (Lecture for Hercules course, available at http://dirac.cnrs-orleans.fr/~kneller/HERCULES/hercules2004.pdf) • Quasi-elastic neutron scattering and molecular dynamics simulation as complementary techniques for studying diffusion in zeolites, H. Jobic and D. N. Theodorou, Micropor. Mesopor. Mater. 102, 21-50 (2007).