1. science
2. physics

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)
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
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.
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/dd
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,)
Gself (r ,0)   (r )
G
self
(r , t )dr  1
I self (Q,0)  1
 0   Sinc (Q,  )d  1
2

Q
1   Sinc (Q,  )d 
2M
Gs(r,0) = (r)
Gs(r,t )  1/V
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 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,).
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) + eu
[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:

Gs (r , t )  D 2Gs (r , t )
t
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:

 Gs (r , t ) 
t
l 2  
6
l2
D
6
 2Gs (r , t )
And we have the following conditions:
Gs (r ,0)   (r )
 G (r, t )dr  1
s
Translational diffusion (Brownian motion)
Gs (r, t )  4Dt 
3 2

exp  r 2 / 4Dt

FT in space

I s (Q, t )  exp  DQ 2t

FT in time
Sinc (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.
(Q)
Sinc (Q,  ) 
  2  (Q) 2
1
1  sin Ql   6 D  sin Ql  
  2 1 

(Q)  1 

Ql  l 
Ql 
3
5



Ql  Ql 


Ql





6D 
3
!
5
!
Ql  1  (Q)  2 1 
  DQ 2
l 
Ql



l =1Å
D = 0.1 Å2meV =
1.519 × 105 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).

1  cosQx a 
(Q ) 
• If crystal oriented with x-axis parallel to Q:
 = 1 meV1 = 0.658 ps
3
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).
Jump model between two equivalent sites
r1

r2

1
1
p(r1 , t )   p(r1 , t )  p (r2 , t )
t



1
1
p (r2 , t )  p (r1 , t )  p (r2 , t )
t



 p(r1 , t )  p(r2 , t )  0
t
Solutions are:
p(r1 , t )  p(r2 , t )  1
And assuming that at t = 0, the atom is at r1:
p(r1 , t )  A  Be 2t /
p(r1 ,0)  A  B  1
p(r2 , t )  A  Be 2t /
p(r2 ,0)  A  B  0
Jump model between two equivalent sites
r1
r2
2 t
1
p(r1 , t ; r1 , 0)  [1  e  ]
2
2 t
1
p (r2 , t ; r2 , 0)  [1  e  ]
2
2 t
1
p (r2 , t ; r1 , 0)  [1  e  ]
2
2 t
1
p (r1 , t ; r2 , 0)  [1  e  ]
2
I (Q, t )  [ p(r1 , t; r1 ,0)  p(r2 , t; r1 ,0)eiQ ( r2 r1 ) ] p( r1,0)
[ p(r1 , t; r2 ,0)eiQ ( r1 r2 )  p(r2 , t; r2 ,0)] p( r2 ,0)
2 t
1
1
I (Q, t )  [1  cos Q.(r2  r1 )]  [1  cos Q.( r2  r1 )]e 
2
2
Jump model between two equivalent sites
r1
d
r2
1
1
1
2
S (Q,  )  [1  cos Q.(r2  r1 )] ( )  [1  cos Q.( r2  r1 )]
2
2
 4   2 2
If powder, average over all possible orientations
1  sin Qd 
1  sin Qd  1
2 1
S (Q,  )  1 
    1 

2
Qd 
2
Qd    2  2 1

S (Q,  )  A0 (Q)    A1 (Q)
EISF
QISF
1
2 1
  2  2 1 2

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 el (Q,  )
EISF (Q)  el
S (Q,  )  S qel (Q,  )
I (Q, )  e


iQr (  ) iQr ( 0)
e
I (Q, )  e

iQr (  )
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 ekT  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-HI.
(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-HO
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 (50-100eV)
MIBEMOL LLB Saclay (85eV)
IRIS at ISIS GB (15eV)
- 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.Rm ) 
EISF   1

 Q.Rm 
R7
R8
R6
2
Rm  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
1 
Sinc (Q,  )   I (Q, t ) R (Q, t ) exp(it ) dt
2 
R(q,t) : resolution function of IN6
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
MSD(t )   bi2 ri (t  t0 )  ri (t0 )
2
Mean square displacement:
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) +
(1A0R) A0L L(T +R) +
A0R (1A0L) L(T +L) +
(1A0R) (1A0L) L(T +R +L)
If R  0 (MD, NMR Imanari 2010) then:
S(Q,)  A0L L (T) + (1A0L) 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).107m2s-1
Ea = 19 ± 2 kJmol-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 (1010 m2s1) 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 Q2 u 2
2
u2
1.0
-1
Q(A )
1.5
2.0
A (Q) (E)  1 A (Q)LW , E  (solid)
A (Q)LW ,E 1 A (Q)LW ,E (liquid)
0
0
0
t
r
0
t r
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 1010 m2/s vs 3.2 x 1010 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).