Quasielastic Neutron Scattering

Miguel A. GonzalezInstitut Laue-Langevin (Grenoble, France)

gonzalezm@ill.eu

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 thescattering 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) !

From Jobic & Theodorou, Micropor. Mesopor. Mater. 102, 21-50 (2007)

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 Heberle et al., Biophys. Chem. 85, 229-248 (2000)

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

PELICAN@ANSTO

A true QEINS spectrum: waterTeixeira et al., Phys. Rev. A 31, 1913 (1985) Qvist et al., J. Chem. Phys. 134, 144508 (2011)

• 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.

IN6@ILL

IN5@ILL

• 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.

Instrumental constraints

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

intermediate scattering function, I(Q,t)

DIRECT RELATION: Measured quantity Physical information

d2/dd S(Q, )

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.

Self correlations (incoherent scattering)

self intermediate function

FT in time FT in space

Sinc(Q, ) Iself(Q,t) Gself(r,t) [energy]1 [] [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,)

Gs(r,0) = (r)

Gs(r,t ) 1/V

)()0,( rr selfG

1),( rr dtGself

1)0,( QselfI1),(0 dSinc Q

M

QdSinc 2

),(2

1 Q

FROM THE GENERAL EXPRESSION TO USEFUL MODELS

Self intermediate scattering function

• 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,).

Approximations or assumptions

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:

Sinc(Q,) = B(Q) + eu2Q2 [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).

Fick’s 2nd law tells how diffusion causes concentration to change with time:

Translational diffusion (Brownian motion)

),(),( 2 trGDtrGt ss

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:

),(

6),( 2

2

trGl

trGt ss

6

2lD

And we have the following conditions: )()0,( rrGs

1),( drtrGs

Translational diffusion (Brownian motion)

DtrDttrGs 4/exp4),( 223

tDQtQI s2exp),(

222

21),(

DQ

DQQSinc

Neutron spectrum is a lorentzian function with a width increasing strongly with Q: HWHM = DQ2.

FT in space

FT in time

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.

22 )(

)(1),(

Q

QQSinc

Ql

Ql

l

D

Ql

QlQ

sin1

6sin1

1)(

2

2

53

2!5!31

6)(1 DQ

Ql

QlQlQl

l

DQQl

l = 1 ÅD = 0.1 Å2meV = 1.519 × 105 cm2/s

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:

3

cos1)(

aQQ x

= 1 meV1 = 0.658 ps

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).

r1 r2

1 21

1 1( , ) ( , ) ( , )p t p t p t

tr r r

2 1 2

1 1( , ) ( , ) ( , )rp t p t p t

tr r

Jump model between two equivalent sites

21( , ) ( , ) 0p t p trrt

21( , ) ( , ) 1p t p r tr

/21 ),( tBeAtrp

/22 ),( tBeAtrp

And assuming that at t = 0, the atom is at r1:

1)0,( 1 BArp

0)0,( 2 BArp

Solutions are:

r1 r2

1 1

21( , ; ,0) [1 ]

2

tp rt er

2 1

21( , ; ,0) [1 ]

2

tp rt er

2 2

21( , ; ,0) [1 ]

2

tp rt er

1 2

21( , ; ,0) [1 ]

2

tp rt er

1 1

2

2 2

1 1( , ) [1 cos .( )] [1 cos .( )]

2 2

tI Q t Q Qr r er r

1

1

2

2

1 1 1( )

(

2

2 2 2 21)

1( , ) [ ( , ; ,0) ( , ; ,0) ] ( ,0)

[ ( , ; ,0) ( , ; ,0)] ( ,0)

iQ

iQ

r

r

r

r

r

r r

I r r

r

Q t p t p t e p

p t e p t p r

r r

r

Jump model between two equivalent sites

r1 r2

Jump model between two equivalent sites

22 1 21 2

1 1 1 2( , ) [1 cos .( )] ( ) [1 cos .( )]

2 2 4r r rS Q rQ Q

If powder, average over all possible orientations

212

1

2

21sin1

2

1sin1

2

1),(

Qd

Qd

Qd

QdQS

d

212

1

102

21)()(),(

QAQAQS

EISF QISF

r1 r2

Jump model between two equivalent sites

d

Half width ~1/(independent of Q)

w0

A0d(w)

r1 r2

Jump model between two equivalent sites

d

Qr

A0(Q) = ½[1+j0(Qd)/(Qd)] EISF

1

½

Q

HWHM

EISFs corresponding to different rotation models

EISFs and widths of different rotation models

Physical meaning of the EISF

),()(),(),( QSQIQS qel ),()( QIQEISF

)0()(),( rQirQi eeQI

And if the system is in equilibrium, there are no correlations between positions at t=0 and t=, so:

And the EISF is easily obtained as the ratio between the elastic intensity and the total (elastic + quasielastic, no DW) intensity:

),(),(

),()(

QSQS

QSQEISF

qelel

el

2)0(

2)()0()(),( rQirQirQirQi eeeeQI

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)

Tunnelling spectrum of NH4ClO4 and with different levels of partial deuteration

Probe potential energy barriers and rotational potentials (test for simulations)

Nuclear hyperfine splitting of Nd

Low-frequency excitations

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

Heuser et al., PRL 2014

- 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.

- DFT shows metastable sites characterized by a lower activation energy for diffusion.

- DPD expected to depend on H concentration and dislocation density.

QENS represent a unique experimental scenario that allows the diffusivity associated to dislocation pipe diffusion to be directly quantified!

(QENS ~ 230 meV)

(QENS ~ 40-80 meV)

- 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.

Rodriguez-Velamazan et al., JACS 2012

HS LS Bz

295 K(IN5)

- 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.

- 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.

Bz (PyH)I EISF (6 equivalent sites)

Pajzderska et al., JCP 2013

(PyH)NO3 @ IN10 (ILL)

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.

QENS MD

Dealing with complex systems …

How does the microscopic structure and dynamics change with varying alkyl chain length?

- In most cases, need some kind of computational model to understand and interpret the QENS spectra.

- 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 …)

reduced oxidized

Polyaniline – Emeraldine base (insulating form)

H+

Emeraldine salt (conducting form)

HN

N N

N

H H

H

++

HN

N N

N

H

Structural Analysis

3.5 Å

~25-38 Å

Structural model

Quasi-elastic neutron scattering (QENS) studies of polyaniline/DB3EPSA

TOF Spectrometers:

IN6 ILL (50-100meV)

MIBEMOL LLB Saclay (85meV)

IRIS at ISIS GB (15meV)DB3EPSA

- 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

msaRRm ,

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!R1

R3R4

R5

R6

R7

R8

R9

R10

Model of local diffusion of protons in spheresVolino and Dianoux, Mol. Phys.41, 271,

(1980)

2

1

.

).(

m

m

RQ

RQjEISF

-0,4 -0,2 0,0 0,2 0,40,0

0,2

0,4

0,6

0,8

1,0

S(q,

)

E (meV)

-1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,00,00

0,02

0,04

0,06

0,08

S

(q,)

E (meV)

q=2 A-1

T=235K simulation experiment

T=280K simulation experiment

T=310K simulation experiment

T=340K simulation experiment

resolution function

dttitQRtQIQSinc )exp(),(),(

21

),( R(q,t) : resolution function of IN6

Dynamic structure factor Sinc(q,w) IN6 spectrometer

R1

R2

R6R10

R0

DB3EPSA

PANI chain

R1R0

R2

R6

R10

T=340 K

i

iii trttrbtMSD 200

2 )()()(

Analysis of the individual atom trajectories

Mean square displacement:

MD confirms the model employed to fit the QENS spectra!

A complex example containing several contributions …

How does the microscopic structure and dynamics change with varying alkyl chain length?

EmimBr or C2mimBr

BmimBr or C4mimBr

HmimBr or C6mimBr

Room temperature ionic liquids based on the imidazolium cation

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-A0

R) L(R)

L(Q, ) A0L + (1-A0

L) L(L)

S(Q,) A0RA0

L L(T) +

(1A0R) A0

L L(T +R) +

A0R (1A0

L) L(T +L) +

(1A0R) (1A0

L) L(T +R +L)

If R 0 (MD, NMR Imanari 2010) then:

S(Q,) A0L L (T) + (1A0

L) L (T +L)

QENS analysis

0 1 2 3 40.00

0.03

0.06

0.09

0.12

Q2(Å2)W

t (m

eV)

354K

374K

393K

412K

T (K) D (1010 m2s1) t0 (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

D follows Arrhenius law: D = D0 exp(Ea / RT) with

D0 = (1.7±0.8).107m2s-1

Ea = 19 ± 2 kJmol-1

Reasonable agreement with NMR (Every, PCCP 2004), although D values 3-4 times larger.

Data fitted with two lorentzians: 1 translational-like + 1 local-like

QENS: C2mimBr translational dynamics

S(Q,E) exp Q2 u2 A0(Q)L W t ,E 1 A0(Q) L W tr,E

S(Q,E) exp Q2 u2 A0(Q)(E) 1 A0(Q) L Wr,E

0.5 1.0 1.5 2.0

0.2

0.4

0.6

0.8

1.0

A0(Q

)

Q(A-1)

250K 300K 354K 374K 393K 412K

(solid)

(liquid)

QENS: C2mimBr local dynamics

1

R2

0.4 0.8 1.2 1.6 2.0 2.4 2.8

0.25

0.50

0.75

1.00

250K 300K 354K 374K 393K 412K

A0(Q

)

Q(Å-1)

EmimBr

160 240 320 400 4800

1

2

3

P

R1

R (

Å)

T(K)

R2

0.2

0.3

0.4

0.5

Pro

bab

ility

(b) Crystal State (c) Liquid State

(c)

(a)

(b)

0.5 A

(d)

R1

R2

2

3

1

4

QENS analysis using MD input (C2mimBr)

Aoun et al. , J. Phys. Chem. Letters 1, 2503 (2010)

Quasielastic widths: Simulation vs experiment

Liquid 360K

Liquid 360K

Crystal 300KSimulation: 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.!

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

CH2 in ethyl chain CH3 in ethyl chain

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!

• 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

CONCLUSIONSOr why should I use Quasi-elastic Neutron Scattering?

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.neutron-sciences.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).