STATISTICAL MOTIONSSTATISTICAL MOTIONS

M. Bée

Institut de Biologie Structurale CEA-CNRS Laboratoire de Biophysique Moléculaire

41 rue J. Horowitz, 38027 Grenoble Cedex-1

Institut Laue-LangevinBP 156, 6 rue J. Horowitz, 38042 Grenoble Cedex-9

Laboratoire de Spectrométrie Physique Université J. Fourier Grenoble 1BP 87 38402 Saint-Martin d’Hères Cedex

M. Bée

Institut de Biologie Structurale CEA-CNRS Laboratoire de Biophysique Moléculaire

41 rue J. Horowitz, 38027 Grenoble Cedex-1

Institut Laue-LangevinBP 156, 6 rue J. Horowitz, 38042 Grenoble Cedex-9

Laboratoire de Spectrométrie Physique Université J. Fourier Grenoble 1BP 87 38402 Saint-Martin d’Hères Cedex

0kkQ

Neutronsincidents(E0, 0, k0)

Détecteur

EF < E0

kF < k0

Perte d’énergie

du neutron

EF > E0

kF > k0

Gain d’énergie

du neutron

EF = E0

kF = k0

Pas d’échange

d’énergie

Q

Q

Q

k0Echantillon

PRINCIPLE OF A NEUTRON EXPERIMENT

ii) Energy transfer, difference between final, E, and initial, E0, neutron energy,

)(2

20

22

0 kkm

EE

i) Momentum transfert, Q, difference between final, k, and initial, k0 neutron momentum

Two relevant quantities are measured:

dtetQ.ri

eQ.ri

ebbNk

k

dd

d ti )(

)0(

.2

11

0

2

Average over spin states Average over positions

Differential scattering cross sections

Thermal average over the positions of the nuclei and over their spin states.In general, can be taken separately.

dtetQ.ri

eQ.ri

ebbNk

k

dd

d ti )(

)0(

2

11

0

2

One component system

cohb b

22 incb b b

2 2

0

22

0

(0) ( )1 1.

2

(0)1 1 ( ). 2

i t

i t

i i td kb e e e dt

d d k N

ik i tb b e e e dtk N

Q.r Q.r

Q.r Q.r

Incoherent scattering length

Coherent scattering length

SCATTERING FUNCTIONS2 2 2

coh inc

d d d

d d d d d d

Coherent scattering Incoherent scattering

Intermediate scattering function

( )1 (0)( , ) .

i tiF t e e

N

Q.rQ.rQ

Spatial Fourier transform of time-dependent pair correlation function

3

1( , )

(2 )iG t F( ,t) e d

Q.rr Q Q

Time Fourier transform: dynamical scattering function (scattering law, dynamical structure factor)

1( , )

2i tS F( ,t) e dt

Q Q

( , ) iF t G( ,t) e d

Q.rQ r r

1 ( , )

2i tF( ,t) S e d

Q Q

( )1( , )

2i tS G( ,t) e d dt

Q.rQ r r( )

3

1( , )

(2 )i tG t S( , ) e d d

Q.rr Q Q

Intermediate incoherent scattering funtion

(0)1 ( )( , ) . inci i tF t e e

N

Q.r Q.rQ

1( , )

2i t

inc incS F ( ,t) e dt

Q Q

Space-time Fourier transform of time dependent autocorrelation function

( )1( , )

2i t

inc sS G ( ,t) e d dt

Q.rQ r r

( )3

1( , )

(2 )i t

s incG t S ( , ) e d d

Q.rr Q Q

INCOHERENT SCATTERINGSame scatterer interacts with neutron at times 0 and t

Incoherent scattering funtion

Finally 2

0

1( , ) ( , )

4 coh inc inc

d kS S

d d k

Q Q

Coherent scatteringcross section 2

4 cohcoh b 2

4 incinc b

Incoherent scatteringcross section

Lattice vibrations Equilibrium position within molecule with orientation Internal molecular vibrations

( ) ( ) ( , ) ( )t t t t r R uv

SEPARATION OF ATOMIC MOTIONS

Instantaneous position of a scattering nucleusBelonging to a reorienting molecule or chemical group:

TWO CASES

No long-range motions (rotator phases in solids, liquid crystals…)

Existence of long-range displacements(liquid, intercalated species…)

( ) ( ) ( , ) ( )t t t t r T R u

C.O.M translation Equilibrium position within molecule with orientation Internal molecular vibrations

Basic assumption: all the components of the atomic displacement are completely decoupled

Occur on different time scales:Lattice vibrations and internal molecular vibrations: 10-13 – 10-14 sMolecular reorientations: 10-11 – 10-12 sTranslation of molecule C.O.M in liquids: 10-9 - 10-10 s

-4

-2

0

2

4

6

8

10

0 20 40 60 80 100

Temps (picosecondes)

xx

y

z

External (lattice)and internalvibrations

Moleculereorientations

Example: molecular dynamic simulation of hexamethylethane (CH3)3-C-C-(CH3)3

( , ) exp . (0) .exp . ( )

exp . (0) .exp . ( )

exp . ,0) .exp . ( , )

incF t i i t

i i t

i i t

Q Q Q

Q u Q u

Q R( Q R

v v

Ω Ω

Intermediate scattering function:No long-range displacements Long-range displacements

( , ) exp . (0) .exp . ( )

exp . (0) .exp . ( )

exp . ,0) .exp . ( , )

incF t i i t

i i t

i i t

Q Q T Q T

Q u Q u

Q R( Q RΩ Ω

( , ) ( , ). ( , ). ( , )inc V U RF t F t F t F tQ Q Q Q ( , ) ( , ). ( , ). ( , )inc T U RF t F t F t F tQ Q Q Q

Incoherent scatteringfunctions

( , ) ( , ) ( , ) ( , )inc U RS S S S Q Q Q QV ( , ) ( , ) ( , ) ( , )inc T U RS S S S Q Q Q Q

Quantum mechanicClassical mechanic

Quantum mechanicClassical mechanic

LATTICE AND MOLECULAR VIBRATIONS

0

0

(0) ( )( , ) .

( , ) exp .exp

. (0) . ( )

inc

t

t

i i tF t e e

F t U U

U i U i t

Q.u Q.uQ

Q

Q u Q u

Operators U0 and Ut do not commutte, but each of them commutte with their commutator [U0, Ut].

0 0 0

20 0

exp( ).exp( ) exp( )exp([ , ])

( , ) exp exp

t t t

t

U U U U U U

F t U U U

Q

2

22

( , ) exp . (0) .exp . (0) . ( )

1exp . (0) . 1 . (0) . ( ) . (0) . ( ) ...

2

F t t

t t

Q Q u Q u Q u

Q u Q u Q u Q u Q u

Purely inelastic term involving in its expansion

the autocorrelation function of the atom

displacement.

Expansion to second order of the second exponential term

2 2 2exp - (0) exp -

exp -2 ( )

Q u

W Q

Q.u

Debye-Waller term: interference between the neutronic wave scattered by atom at times 0 and t is attenuated

by its own thermal vibrations

( , ) exp 2 ( ) ( ) ( , ) inélS W Q S Q Qv v v

0Energy exchange

Example: Methyl torsion

30 meV

INTERNAL VIBRATIONS

Debye Waller term 2 22 ( )W Q Q v v

( , ) exp 2 ( ) ( ) ( , ) inélu u uS W Q S Q Q

0

Acoustic modesBrillouin zone limits

Optical modes:Brillouin zone limitsCentre of Brillouin zone

5 –10 meV

Debye Waller term

Energy exchange

EXTERNAL VIBRATIONS

2 22 ( )W Q u Q u

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0 5 10 15 20 25 30 35 40

Vib

ratio

nal D

ensi

ty o

f S

tate

s (A

rb. u

nits

)

Energy (meV)

0

0.0001

0.0002

0.0003

0.0004

0.0005

0 5 10 15 20 25 30 35 40

Vib

ratio

nal D

ensi

ty o

f S

tate

s (A

rb. u

nits

)

Energy (meV)

(a)

(b)

100 K

271 K

235 K

185 K

300 K320 K

2 = 95°

2 = 25°

2 = 48°

Example 2: camphor sulfonic acid

0 50 100 150 200 250

Nombre d'onde (cm-1)

0 500 1000 1500 2000

0

Inte

nsité

(un

ité a

rbitr

aire

)

Energie (meV)

Example 1: Cytidine T = 15 KGaigeot, M.-P., Leulliot, N., Ghomi, M., Jobic, H. Coulombeau, C. Bouloussa, O.,Chemical Physics 261 (2000) 217

CASE OF A MOLECULE

3N - 6 harmonic oscillators

N atoms: 3N – 6 frequencies 3N – 6 Dirac peaks

Time of flight(10-11-10-12 s)

Multichoppers

IN5 ILL GrenobleMibemol LLB SaclayNEAT HMI BerlinDCS NIST USA

Crystal monochromator

IN6 ILL GrenobleFOCUS PSI SwitzerlandFCS NIST USA

Inverted TOF geometry

IRIS RAL UK

Backscattering(10-9-10-10 s)

Doppler effect

IN10 ILL GrenobleBSS KFA Jülich IN16 ILL GrenobleHFBS NIST USAFRMII (project) Munich

Thermal expansion

IN13 ILL Grenoble

Spin echo(10-8-10-9 s)

Standard

IN11 ILL GrenobleMESS LLB SaclayIN15 ILL Grenoble

Zero field

HMI BerlinLLB Saclay

OVERVIEW OF SPECTROMETERS FOR QENS

PROPERTIES OF TRANSLATIONAL AND ROTATIONALSCATTERING FUNCTIONS

( , ) (0); ( ) . (0) .exp . (0) .exp . ( ) . ( ). (0)incF t p t p i i t d t d Q R R R Q R Q R R R

)();0( tRRp Probability for a scattering atom, located at R(0) at initial time, to be at R(t) at time t

)0(Rp Distribution of probabilities at initial time

( (0); ( )) ( (0)) if p R R p R t

( , ) ( ) (0) exp . (0) exp . ( ) ( ) (0)

(0) exp . (0) (0). ( ) exp . ( ) ( )

incF p p i i d d

p i d p i d

Q R R Q R Q R R R

R Q R R R Q R R

Return to equilibrium:

System in equilibrium ( ( )) ( (0)) p R p R

2

2

( , ) (0) exp . (0) (0)

( ) exp . ( ) ( )

incF p i d

p i d

Q R Q R R

R Q R R

2( , ) exp( . (0))incF Q iQ R

At any arbitrary time, formal separation

time independing part

( , ) ( , ) ' ( , )inc inc incF t F F t Q Q Q

Time-Fourier transformation

Purely elastic component

1 1( , ) ( , ) exp( ) ' ( , ) exp( )

2 21

( , ). ( ) ' ( , ) exp( )2

inc inc

inc inc

S F i t dt F t i t dt

F F t i t dt

Q Q Q

Q Q

' ( , ) ( ,0) ( , ) .expinc inc inct

F t F F Q Q Q

2 2

( , ) ( , ). ( )

1( ,0) ( , . .

1

inc

inc inc

S F

F F

Q Q

Q Q

time depending part

Inelastic component the shape and width of whichdepend on the nature and of the characteristic timesassociated to nucleus motions

Very simple case: exponential decreasing, with a unique characteristic time,

Temps

Iinc

(Q,t)

(Iinc

(Q,0)-Iinc

(Q,)).exp(-t/

Iinc

(Q,0)

Iinc

(Q,

0

5

10

15

20

25

30

-600 -400 -200 0 200 400 600

Echange d'énergie du neutron (eV)

2/

Elastic Incoherent

Structure Factor (EISF)

2( ) ( , ) exp( . ( )

( )

( ) ( )

inc

EL

EL QEL

EISF Q F Q iQ R

Int Q

Int Q Int Q

Immediate information about motion geometry.Vanishing in the case of long-range diffusive motions (liquids).For localised motions, the more the proton distribution is isotropic in space,the weaker is the elastic contribution to scattered intensity at large momentum transfer.

LONG RANGE DIFFUSION

2 2 22

2 2 2

( , ) ( , ) ( , ) ( , )( , )

c t c t c t c tD c t D

t x y z

r r r rr

c(r, t) concentration of diffusing atomsD, diffusion coefficient (macroscopic)

( , )( , )s

c tG t

N

rr

Fick’s equation is also fulfilled by Gs(r, t ). Solution: normalised gaussian function

Phenomenological equation: second law of Fick (1855)

System at equilibrium: van Hove autocorrelation function,

Gs(r, t) = probability for an atom, initially located at origin,to be at r at time t:

N = total number of atoms

2

3/ 2

1( , ) exp

(4 ) 4s

rG t

Dt Dt

r Initial condition: Gs(r, t) = δ(r).

Function isotropic in space (no driving force in any direction)Mean square displacement of an atom proportional to time

2 ( ) 6r t Dt

Corresponds to the microscopic description of Einstein’s random walk modelEinstein (1905), Smoluchowski

a

TkD B

6

Translational diffusion coefficient:Brownian tracer sphere with radius a in a fluid with shear viscosity Brownian motion (1827)

Can be still valid when sizes of tracer and solvent molecules are comparable

Earlier QENS experiments (30 years ago) mainly devoted to studies of liquidsEarlier QENS experiments (30 years ago) mainly devoted to studies of liquids

Second Fick’s law

2 2 22

2 2 2

( , ) ( , ) ( , ) ( , )( , )

c t c t c t c tD c t D

t x y z

r r r rr

Spatial Fourier transform intermediate scattering function

2( , ) ( , ) exp( )F t F Q t DQ t Q

Gaussian function isotropic with Q

Remark:

lim ( , ) 0t

F Q t

Any atom can move away from origin from an arbitrary distance, the EISF is zero

Spatial Fourier transform dynamical structure factor

Lorentzian function, isotropic with Q Halfwidth at half maximum, , increases as a function of momentum transferaccording to a parabolic law

2

22 2

1( , )inc

DQS Q

DQ

SCATTERING LAW; DYNAMICAL STRUCTURE FACTORH

alw

idth

at h

alf

maxi

mu

m

Q2

HWHM = DQ2

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0 0,05 0,1 0,15 0,2 0,25 0,3

Liquid paraxylene

Hal

f Wid

th a

t H

alf M

axim

um (

meV

)

Q2(A-2)

DQ2 law

Earlier studies:Liquid argon

Dasannacharya and RaoPhys. Rev. 137 (1965) A417

Coherent scattering:de Gennes narrowing

More recently observedfor liquid paraxylene

Jobic et al. (2001)

Study of liquid waterJ. Texeira et al. J. Phys. Coll. 45, C7 (1982) 65J. Texeira et al. Phys. Rev. A 31 (1985) 1913

Description with the modelOf Singwi and Sjölander

DQ2 law actually verified at small Q values

Correspond to large distances (several tens de of molecular diameters).

At large Q values, systematic deviations: over small distances, microscopic mechanisms intervene.

JUMP DIFFUSION

Deviation with respect to Fick’s Discontinuous character of diffusion mechanism.

Different models: application to different systems Hydrogen in metals Molecules adsorbed inside porous media (zeolites) Molecular liquids (water)

The model of Chudley and Elliott Initially developped for liquids with a short-range order.Numerous applications (e.g. diffusion of hydrogen atoms adsorbed in metals)

Hypotheses:

Vibrations about an average position (site) during ,Displacement towards another site by instantaneous jumps,Jump distance, l, >> vibrational displacements, Equilibrium sites on a Bravais lattice.

1

( , ) 1( , ) ( , )

n

ii

P tP t P t

t n

r

r l r

Diffusion equation (master equation) Analogous to Fick’s equation

P(r, t): probability for one atom to occupy a given siteSum over the n nearest neighbour sites located at distances li.

Fourier transformation: intermediate scattering function F(Q, t)

Differential equation

1

( , ) 11 exp( . ) ( , )

n

i

F ti F t

t n

i

QQ l Q

Solution

1

1( , ) exp 1 exp( . )

n

i

F t i tn

iQ Q l

Intermediate scattering function non isotropic: depends on the orientation of scattering vector Q with respect to cristallographic directions.

22

1 ( )( , )

( )incS

Q

QQ

Dynamical structure factor

Halfwidth at half-maximum

1

1( ) 1 exp( . )

n

i

in

iQ Q l

DEPENDS ON THE DIRECTION OF THE SCATTERING VECTOR Q

Remarks:

Expressions more complicated in the case of jumps towards sites of different types involving several jump probabilities,Dependence with the direction de Q enables to discriminate between several hypotheses of dynamical models

Case of powders Average over all the orientations of the crystallites with respect to Q

No analytical expression . Numerical average.

Sum of Lorentzian functions with diferent widths.Low values of Q: Sinc(Q, ) Lorentzian shapeLarge values of Q: clear differences.

Jumps between octahedral sites

Distance 3 Å

-1.0 -.5 0.0 .5 1.0

Inte

nsité

(u

.a.)

0

1

2

3

4

-1.0 -.5 0.0 .5 1.0

0

1

2

3

Energie

-1.0 -.5 0.0 .5 1.0

0

2

4

Q = 1 Å-1

Q = 2 Å-1

Q = 3 Å-1

Evaluated

profile Lorentzian

Q (Å-1)

0 1 2 3 4

HW

HM

0.00

.05

.10

.15

.20

.25

.30

Halfwidth of the average

of theLorentzianfunctions

Chudley and Elliott

0

1 1( ) 1 exp( cos )sin

2

1 sin( )1 )

Q iQl d

Ql

Ql

Noticeably differs from the correct result (from the halfwidth of the average of the lorentzian functions)

Chudley and Elliott suggest an averageof the halfwidth at half maximum over all directions of Q

Q2 (Å-2)0 1 2 3 4

HW

HM

0

1

2

3

4

5

6

Q2 (Å-2)

0.0 .2 .4 .6 .8

HW

HM

0

1

2

3

4

5

6

(a)

(b)

At large Q values, oscillatory term negligeable

1lim ( )Q

Q

2 2

0

2 2

1lim ( ) 1 1 ...

3!

6

Q

Q lQ

Q l

Einstein relation

At small Q values, development of the sine function

At low values of Q, Chudley-Elliott model

DQ2 law

Diffusion constant, D, related to residence time t,

2

6

lD

1 sin( )( ) 1 ( ) )

QlQ g l dl

Ql

Other models

Chudley-Elliott model can be generalised:distribution of jump distances, g(l)

Hall and Ross: 2 2

2300

2( ) exp

22

l lg l

ll

2 2

01( ) 1 exp

2

Q lQ

Singwi and Sjölander:

succession of vibratory and oscillatorystates (water).

Chudley-Elliott delocalisation Singwi-Sjölander

r (Å)

0 5 10 15 20 25 30

(r)

0.00

.02

.04

.06

.08

.10

.12

Jobic: delocalisation of the molecules around their sites.

Chudley Elliott

Hall- Ross

Singwi-Sjolander

Hydrogen diffusion

Dominant role within the past (inc(H) = 80 10-24 cm-2)

Still very active field

Hydrogen in TaV5

A.V. Skripov et al.1996

Proton diffusion in relation with protonic conductivity

Perovskites aliovalently doped or non with stochiometric composition: At high temperature: Oxygen ionic conductors

In moist atmosphere can dissolve several mol% of water: proton conductivity

Grotthuss mechanism: proton exchange along an hydrogen bond + OH rotation

Yb-doped SrCeO3 in the temperature range 400-1000°C

(Hempelmann et al., 1995; Karmonik et al,. 1995) Water-doped Ba[Ca0.39Nb0.61]O2.91 between 460 and 700 K. (Pionke et al., 1997)

Systems with hydrogen bonds in stoichiometric quantityHigh conductivity in high temperature phases Several energetically equivalent sites available for each protonCsHSO4 (Colomban et al. 1987; Belushkin et al. 1992; Belushkin et al. 1994)CsOH.H2O (Lechner et al. 1991; Lechner et al. 1993)

ROTATION

EXCHANGE

Solid ionic conductors

Activity has grown rapidly because of potential applications:batteries, fuel cells, electrochemical and photoelectrochemical devices

REQUIREMENTS FOR QENS Favourable neutron cross sections, Ions with high ionic mobility small ionic radius single charge (H+, Li+, Na+, Ag+ or OH-, F- and Cl-)

Ag+ diffusivity in AgI, Ag2Se, Ag2Se, and Rb4Ag4I5

(Funke 1987, 1989; 1993, Funke et al. 1996, Hamilton et al. 2001)

Na+ motions in the two-dimensional ionic conductor -Al203

(Lucazeau et al. 1987) Na+ motions sodium silicate glass, Na2O.2 SiO2

(Hempelmann et al. 1994)

Li+ cations in antifluorite Li2S: diffusion via interstitial hopping

(Altorfer et al. 1994a) Li+ in Li12C60 fulleride (Cristofolini et al., 2000)

Cl- diffusion in SrCl2 and yttrium doped single crystal (Sr,Y)Cl2.03 (Goff et al. 1992)

Intermetallic alloys

Applications for high-temperature environments (aerospace industry)

Diffusion of the constituents themselves

Knowledge of the kinetics on the atomic scale permits:

to control industrial crucial processesto optimize macroscopic characteristics

CsCl structures: NiGa CoGaM. Kaisermayr et al. Phys. Rev. B 61 12038 (2000)

Mechanism of self diffusion not yet fully understoodStructure can host a large amount of vacanciesJump to vacant sites

1- Nearest Neighbour ? Disturbs local order Shorter jump distance

2- Next Nearest Neighbour ? Same sublattice Longer jump distance

E ( µ e V )

- 4 - 2 0 2 4

S(Q,

) (arb. units)

0,0 0,5 1,0 1,5 2,00,0

0,1

0,2

0,3

Two distinct time-scalesLong residence time on regular latticeSmall width

Short residence time on antistructure sitesLarge width

IN16

Quasielastic width

Zeolite MFI structure (silicalite, ZSM-5)

Unit cell: (a = 20.07 Å, b = 19.92 Å, c = 13.42 Å) 96 tetrahedral units, • Si or Al as central atom• Oxygen as corner atoms•4 straight channels•4 zigzag channels•4 channel intersections•10-membered oxygen rings (Ø ~ 5.5 Å)

LONG RANGE DIFFUSION IN MICROPOROUS MATERIALS

Confined media: many effects can be observed: finite size effects, interface effects, dimensionality, liquid dynamics, thermodynamics

Challenge for the theoreticiansImportant industrial processes:• catalysis,• separation of gases and liquidsWide variety of experimental techniques:

Results can vary by up to six ordersof magnitude• Differences between microscopic and macroscopic • QENS and pulsed-field gradient NMR complementary

Workshop« Dynamics in confinement »

Jan 26-29, 2000 GrenobleB. Brick, R. Zorn, H. Büttner

J.Physique IV, Vol 10 PR7-2000

E (µeV)

-10 -5 0 5 10

Inte

nsity

(a.

u.)

0

100

200

300

400

-10 -5 0 5 10

Inte

nsity

(a.

u.)

100

200

300

400

500

n-octaneT = 400 K

isobutaneT = 570 K

Q = 0.87 Å-1

J. Phys. Chem. B 103 (1999) 1096

IN16

Q = 0.87 Å-1

n – octane

T = 400K

Isobutane

T = 570K

H. Jobic et al.J. Phys. Chem. B 103 (1999) 1096

Number of carbon atoms

0 2 4 6 8 10 12 14 16 18

D (

cm2 s-1

)

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

QENS

hierarchicalsimulation

MD

single-crystal

ZLC

linear alkanes / MFI (T = 300 K)

J. Mol. Catal. A 158 (2000) 135

Deviations increase with the number of carbon atoms along the chain:

MD: infinite dilution, rigid framework, no thermalisationBetter agreement at higher loading

Kinetic separation of linear/branched alkanes in Zeolite MFI structure (silicalite, ZSM-5)

time (ns)

0.1 1 10

I(Q,t)

/I(Q

,0)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

C6D6 / ZSM-5 (Q = 0.4 Å-1)

655 K

300 K

465 K555 K

605K

1000/T (K)

1.5 2.0 2.5 3.0 3.5D

(m

2 s-1)

10-16

10-15

10-14

10-13

10-12

10-11

Shah et al.Doelle et al.Zikanova et al. Förste et al.Beschmann et al.Ruthven et al. Rees et al. Karge & NieenSnurr et al. Forester & SmithNSE (this work)

Benzene / MFI

silicalite (˜ , ¿ , q ) ZSM-5 (™, ¯ , s )

J. Phys. Chem. B 104 (2000) 7130H. Jobic et al.

J. Phys. Chem. B 104 (2000) 2878

Benzene / ZSM5 (NSE)

Time-scale extended to longer timesFlux: high loadingAnalysis at low Q

Deuterated samples required

-0.2 -0.1 0.0 0.1 0.2

0

10

20

30

40

50

-0.2 -0.1 0.0 0.1 0.2

0

1

2

3

CH4 /

AlPO4-5

CH4 /

ZSM-48

E (meV)

Inte

nsity

(a.

u.)

Diffusion in 1D channel systems

Resolution: FWHM: 18 µeV ; Q = 0.3 Å-1

D = 10-8

m2/s

F = 10-11

m2/s1/2

CH4 molecules can cross each other

CH4 molecules cannotcross each other

tDx 2 2

tFx 2 2

LOCALISED MOTIONS Jump over two inequivalent sites

11

1 2

11 2

( , ) 1 1( , ) ( , )

( , ) 1 1 ( , ) ( , )

dp tp t p t

dt

dp tp t p t

dt

rr r

rr r

2

22

Equilibrium solution

11 1

1 2

22 2

1 2

( ,0) ( , )

( ,0) ( , )

p p

p p

r r

r r

Solutions:

2

1

( , ) exp( / )

( , ) exp( / )

p t A B t

p t A B t

1

2

r

r

Characteristic time, defined as

1 2

1 1 1

1 2

1

1

2

1

Constant values A and B to be determined from initial conditions

Particle initially on site 1 Particle initiallly on site 2

1 1

22 1

1

( ,0; ,0) 1

( ,0; ,0) 0

p A B

p A B

r r

r r

1 2

22 1

1

( ,0; ,0) 0

( ,0; ,0) 1

p A B

p A B

r r

r r

Conditional probabilities

1 21 1

1 2 1 2

2 22 1

1 2 1 2

( , ; ,0) exp( / )

( , ; ,0) exp( / )

p t r t

p t r t

r

r

1 11 2

1 2 1 2

2 12 2

1 2 1 2

( , ; ,0) exp( / )

( , ; ,0) exp( / )

p t r t

p t r t

r

r

Normalisation1 1 1

1 2 2

( , ; ,0) ( , ; ,0) 1

( , ; ,0) ( , ; ,0) 1

p t p t

p t p t

2

2

r r r r

r r r r

At infinite time, these solutions yield to equilibrium expressions:

11 2 1 1 1

1 2

22 1 2 2 2

1 2

lim ( , ; ,0) lim ( , ; ,0) ( , )

lim ( , ; ,0) lim ( , ; ,0) ( , )

t t

t t

p t r p t r p

p t r p t r p

r r r

r r r

Particle has forgottenits history.

At any arbitrary time

1 1 1 1 1 2 2

2 2 1 1 2 2 2

( , ) ( , ; ,0) ( ,0) ( , ; ,0) ( ,0)

( , ) ( , ; ,0) ( ,0) ( , ; ,0) ( ,0)

p t p t p p t p

p t p t p p t p

r r r r r r r

r r r r r r r

1 1 2 2 1 11

1 2 1 2 1 2 1 2 1 2 1 2

1 2 2 2 2 1

1 2 1 2 1 2 1 2 1 2 1 2

( , ) exp( / ) exp( / )

( , ) exp( / ) exp( / )

p t t t

p t t t

2

r

r

11

1 2

22

1 2

( , )

( , )

p t

p t

r

r

1( , ) ( , ) 1p t p t 2r r

The system is always in equilibrium.

Intermediate scattering function

1 1 2 1 2 1 1

1 2 1 2 2 2 2

( , ) ( , ; ,0) ( , ; ,0)exp .( ) ( ,0)

( , ; ,0)exp .( ) ( , ; ,0) ( ,0)

F t p t p t iQ p

p t iQ p t p

Q r r r r r r r

r r r r r r r

1 2 2 2 12 1

1 2 1 2 1 2 1 2 1 2

1 1 2 1 21 2

1 2 1 2 1 2 1 2 1 2

( , ) exp( / ) exp( / ) exp .( )

exp( / ) exp .( ) exp( / )

F t t t i

t i t

Q Q r r

Q r r

2 2 1 2

1 2 1 22 2

1 2 1 2

21( , ) 2 cos( ) 1 cos( ) exp( / )F t t

Q Q.r Q.r

Depends on the orientation of Q = Q() with respect to the jump vector r = r1 – r2

( , ) ( , )sin F Q t F t d d QPolycrystalline sample

At contrast with the case of long range jump diffusion over absorption sites of a lattice , the exponential term does not depend on the scattering vector Q The powder average yields an analytical expression with a unique exponential function

Powder average

2 2 1 2

1 2 1 22 2

1 2 1 2

21 sin( . ) sin( . )( , ) 2 1 exp( / )

. .

Q r Q rF Q t t

Q r Q r

Term independent on time:localisation of the particle in space

2 2

1 2 1 22

1 2

lim ( , ) ( , )

1 sin( . )2

.

tF Q t F Q

Q r

Q r

Limit at infinite time, of theintermediate scattering function

ELASTIC INCOHERENTSTRUCTURE FACTOR (EISF)

Term dependent on time:Exponential decay of fluctuations

2 21 2

2

1 2

lim ( , )Q

F Q

Oscillatory, with pseudo-période Q.rLimit at high Q

2 21 2

2

1 2

Two sites energetically equivalents

min

min min

min

1 sin( . ) 1 sin( . )( , ) 1 1 exp( / )

2 . 2 .

1 sin( . )lim ( , ) ( , ) 1

2 .

1lim ( , )

2

t

Q

Q r Q rF Q t t

Q r Q r

Q rF Q t F Q

Q r

F Q

If the particle stays indefinitely ober one of the sites whatever it is

2 2 2 21 2 1 2

2 2

1 2 1 21 2lim lim 1

Fourier transform dynamical scattering function

2 21 2 1 22

1 2

1 22 2 2

1 2

1 sin( . )( , ) 2 ( )

.

2 sin( . ) 11

. 1

Q rS Q

Q r

Q r

Q r

32 2

1 sin( 3 2 sin( 3 1( , ) 1 2 ( ) 1

3 33 3 1sites Qr Qr

S QQr Qr

JUMPS OVER 3 EQUIVALENT SITESPolycristalline sample

Example: methyl groups of monohydrated camphor sulfonic acid, C10H15O4S

- - H3O+ (CSA)

3

2 2

12 6( , ) ( ) ( , )

18 18

1 sin( 37 2 ( )

9 3

2 sin( 3 11

9 3 1

sitesS Q S Q

Qr

Qr

Qr

Qr

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 1 2 3 4 5

Sauts de 120° T = 200 KT = 225 KT = 250 KEchange de protonE

last

ic I

nco

her

ent S

tru

ctu

re F

acto

r

Q (A-1)

Limit of IN6 and IN10

Analysis from 15 to 340 K, with backscattering and time of flight

spectrometries. Uncertainties over experimental

points especially high. Low neutron flux

Ratio mobile/total hydrogen atoms 6:18

0

0.01

0.02

0.03

0.04

0.05

1.5 2 2.5 3 3.5 4 4.5 5

T = 200 K

Q (Å-1)

IN13

0

0.0002

0.0004

0.0006

0.0008

0.001

1 1.2 1.4 1.6 1.8 2

T = 150 K

Jum

p-ra

te (

meV

)Q (Å-1)

IN100.02

0.04

0.06

0.08

0.1

1 1.5 2

Q (Å-1)

IN6 T = 300 K

IN10 T = 150 K

IN13 T = 200 K

Halfwidth at half maximumof quasielastic part of spectra:

HWHM is constant with Q. 1 single lorentzian (2 or 3 sites)

CONTINUOUS ROTATIONAL DIFFUSION ON A CIRCLE

),,(.),,( 02

2

0 tPDtPt R

) exp( )(exp2

1),,( 2

00 tnDtintP R

n

0000 )(),( )0,(),(exp),( ddPtPtitI Rinc RRQ.Q

m

RmRinc tmDQRJtI ) exp()sin(),( 22Q

221

01

1 )( )( )(),(

m

m

m

mAAS

QQQ

with the EISF )sin()( 200 QRJA Q

and quasielastic incoherent structure factors

)sin(2)( 2 QRJA mm Q

Rm Dm21 correlation times tm

rotational diffusion coefficient DR

ISOTROPIC ROTATIONAL DIFFUSION OF AN ATOM ON A SPHERE

),,(.),,( 00 tPDtPt R

1

220 )1(exp )( )12( )(),(

l

RlRinc tDllQRjlQRjtQI

221

01

1 )( )( )(),(

l

l

l

l QAQAQS

)()( 200 QRjQA

Coefficients Al(Q) with l  0 are the quasielastic incoherent structure factors

)()12()( 2 QRjlQA ll

The hwhm of the successive Lorentzian functions in the expansion are directly related to the rotational diffusion coefficient DR and increase with l according to

Rl Dll )1(1

The norbornane molecule

The OPCTS molecule

ELASTIC SCANS; THE FIXED-WINDOW METHOD

Quasielastic broadening

Rapid inspection of dynamicsas a function of temperature.

To be completed by quasielastic measurements

Decrease of theobserved elastic intensity.

Backscattering spectrometersIncident neutrons with exactly the same energy as selected by analyseurs.Recording of elastic scatteringVariation of an external parameter (temperature, sometime pressure.)Apparition of a motion in the experimental time-range

Ne

utro

n co

unt

s

Energy transfer

2 2( , 0) exp( )steS Q C Q u

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50 100 150 200 250 300 350

T(K)

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50 100 150 200 250 300 350

Cou

nts

(No

rma

lise

d U

nits

)

2 = 29°

2 = 93°

2 = 156°

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50 100 150 200 250 300 350

T(K)

Diffusive motions too slowHarmonic vibrations<u2(T)> linear with T

Quasielastic broadening too largeHarmonic vibrations<u2(T)> linear with T

<u2(T)> accounts for bothdiffusive and et vibratory motions

2 2( , 0) exp( )steS Q C Q u

2 2( , 0) exp( )steS Q C Q u

Possible extraction of EISF

Example: camphor sulfonic acid

2 2( , 0) exp( )S Q Q v

Example of application: protein dynamicsDynamical transition of myoglobin hydrated with D2O.From 4 K to 180 K elastic intensity varieswith Q according to a Gaussian law For an harmonic solid: Debye-Waller factor

Mean square displacement <v2> proportional to temperature.Near 200 K apparition of new degrees of freedom. Deviation from Gaussian behaviour which increases with temperature

2 22 2

1 2 1 22

1 2

exp( ) sin( . )( , 0) 2

.

Q Q rS Q

Q r

v

Initial slope of the curve.

22 1 2

2 21 20

ln ( ,0)

3( ) ( )Q

d S Q r

d Q

v

From refinement to experimental data,<v2>, r2 et 1/2.

Free energy G1

2exp( / )G RT

Myoglobin (Doster et al.):asymmetry of well DH = 12 kJ.mol-1

Entropy DS/R = 3,0,Jump distance r = 1.5 Å.

EXAMPLES OF ROTATIONAL DIFFUSION OF MOLECULES AND CHEMICAL GROUPS

Mostly based on jump models

Earlier studies: SH reorientations in NaSH, CsSH

Rowe et al., J. Chem. Phys. 58, 5463 (1973)120° jumps of methyl groups in para-azoxy-anisole

Hervet et al. (1976)Rotations of aromatic rings

Chhor et al. (1982)Several axes: use of group theory

Rigny Physica 59 707 (1972)Thibaudier and Volino Mol. Phys. 26 1281 (1973); ibid. 30 1159

(1975)

Some cases of rotational diffusion

OctaphenylcyclotetrasiloxaneBée et al. J. de Chimie Physique 83 10 (1984)

NorbornaneBée et al. Mol. Phys. 51 2 221(1986)

NH NH NH NHn

NH NH Nn

N

DopingAH

Semi-oxydized, non conductive form: emeraldine base

Insoluble powder, non conductive10-10 S/cm

Insoluble powder,1-20 S/cm

Conductive salt of emeraldine

SO3HO

Doping by HCl, H2SO4…

Protonation with a functionalized organicacid: camphor sulfonicacid (CSA)Polar solvent: meta-cresol

Brittle films100-400 S/cm

+

A- A-+

Dynamics of counter-ions in conducting polyaniline.

D. Djurado et al. Phys. Rev. B 65 (2002) 184202

Polyaniline chain

Camphor sulfonic acid

150

200

250

300

350

0 50 100 150 200 250 300

Co

nd

uctivity

(S

/cm

)

Temperature (K)

Electrical transition

Semiconductingregime

Metallicregime

Chains are immobileMotions of CSA

Methyl groups

WholeCSA

DEHEPSA1,2 di(2-ethylhexyl)esterof 4-sulphophtalic acid

DPEPSAdi-n-pentyl ester of 4-sulphopthalic acid

O

S

CH

New families of plastdopants: high conductivity, improved flexibility

B. Dufour et al. Chem. Mat. 13, 4032 (2001).

0.01

0 50 100 150 200 250 300 350

Ela

sti

c In

ten

sit

y

T (K)

0 50 100 150 200 250 300 350

0.01

Ela

sti

c In

ten

sit

y

T (K)

0.001

0.01

0 50 100 150 200 250 300 350

Ela

sti

c In

ten

sit

y

T (K)

0 50 100 150 200 250 300 350

0.01

Ela

sti

c In

ten

sit

y

T (K)

2 = 24.5°

Q = 1.2 A-1

2 = 45.6°

Q = 2.8 A-1

2 = 66.8°

Q = 3.1 A-1

2 = 88.0°

Q = 3.9 A-1

PANI / DEHEPSA : IN13 10-10 s time-scale

0.7

0.75

0.8

0.85

0.9

0.95

1

0 0.5 1 1.5 2

T = 330 KT = 302 KT = 281 KT = 251 KT = 331 KT = 301 KT = 281 KT = 251 K

Ela

stic

Inco

here

nt S

truc

ture

Fa

ctor

Q (A-1)

DPEPSADEHEPSA

EISF is temperature dependent:Variation with T- Of the region of space accessible to scatterers- Of the fraction of mobile scatterers

Given T, DPEPSA more ordered than DEHEPSA - Smaller chain- Linear chain: packing favored

High values of EISFFraction of

non mobile scatterers

20 10

( )( ) 3

j QRA Q

QR

Model: Diffusion inside a sphereF. Volino and A.J. Dianoux; Molecular Physics 41 (1980) 271

00 2 2

, 0,0

1( , ) ( ) (2 1) ( )

( )

ll nn l

l m n

DS Q A Q l A Q

D

22

212 2 2

2 22 2

2 2

6 . ( ) . ( )( ) if

( 1)

3 ( 1)( ) ( ). if

2

nl nl l ln ln n

l l

l nn l l

R QR j QR l j QRA Q Q

R l l R Q

Q R l lA Q j QR Q

Q R

EISF

Structure factors

Parameter # 4

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20

Incomplete Gamma function

(a,x)

x

a = 7

a = 1

a = 4. ( , )mR R a mp

1 2( , ) . ( ) . ( ) ( , ; )m mm

S Q S Q R Polymer chain Counter-ion head Counter-ion tail

Distribution of radii alongcounter-ion tail

3 ParameterspR1

R2

R3

Biological studiesProteins exhibits a complex structure:

several thousand atomsfolds into a unique 3-D structure

Unfolding of biomolecules Loose of biological activity

Biological function Flexibility

Rich dynamical spectrum: 10-14 to 101 s: manyinstruments concerned

Nanosecond - picosecond regime: major role for intermolecular recognitionand for enzymatic reactions: all QENS instruments are concerned

RigidityPrerequisite of

specificitycompromise

FlexibilityProper function

of protein

Non exchangeable H atoms, distributed homogeneously within the samplegive information on the overall dynamics and on those of the larger groups to which they are bound. (J. C. Smith Quarterly Review of Biophys. J. 76, 1043 (1999)

Flexibility of DNA as a function of hydrationA.P. Sokolov et al. J. Biol. Physics 26 S1 (2000)

Dynamic transition at T ~ 200 - 230 KNot clearly understoodImportant for protein function

Slow relaxationprocess

Fast relaxationprocess

At low temperaturelarger flexibility forlow water content

At high temperature a slow relaxation process appears:Cooperative motion of many DNA’s monomers involving all parts of the molecule (backbone and base pairs)

Strongly depends on the level of hydration

Influence of the solvent in biopolymer dynamics

DEVELOPMENT OF THEORETICAL MODELS

Long range diffusionLattice model with gaussian statistics F. Volino et al. Europhys. J. B 16, 25 (2000)Single-file diffusion K. Hahn et al. Phys. Rev. E 59 6662 (1999) K. Hahn et al. J. Phys. A 28 3061 (1995)Extended Hall and Ross model H. Jobic J. de Physique IV 10 (2000) 77

PolymersMode Coupling TheoryRotation-Rate Distribution Model A.Chahid et al. Macromolecules 17 3282 (1994)

BiologyForce Constant Analysis D. Bicout and G. Zaccaï Biophys. J. 80, 1115 (2001)Combined dynamics: jump and diffusion inside a sphere D. Bicout, Proceedings ILL Millenium Symposium (2001)Diffusion inside two concentric spheres D. Bicout Physical Review E, 62, 1, 261 (2000)Influence of environment fluctuations D. Bicout Physical Review E, 64, (2000)Damped collective vibrations, diffusion in anharmonic potential G. Kneller Chemical Physics 261 1 (2000) K. Hinsen et al. Chemical Physics 261 25 (2000)

MOLECULAR DYNAMIC SIMULATIONS

Seem very promissing for futureStill require long computing time.Up to now application to QENS restricted to small moleculesMethods have been developed for larger molecules K. Hinsen et al. Chemical Physics 261 25 (2000) K. Hinsen and G. Kneller J. Chem. Phys. 111, 24 (1999) 10766

Require a reliable interaction potential Can be a serious difficulty (delocalised charges) Can require long preliminary calculations calculations Need to be checked carefully by comparison with experimental results Necessary to evaluate QENS spectra (nMOLDYN) G. Kneller et al. Comp. Phys. Comm. 91 (1995) 191

Under that conditions, analysis of atoms trajectories can be of considerable help in understanding neutron resultsDistribution of Potential barriers for methyl groups in glassy polyisopreneF. Alvarez et al. Macromolecules 33 (2000) 8077

J. Combet; Journées de la diffusion neutronique 2001

Orientational distribution

FCC PhaseFCC Phase

Dynamics more complexThan an isotropic rotation(preferential orientations).

HCP PhaseHCP Phase

MOLECULAR DYNAMIC SIMULATION OF NORBORNANE

Good agreement with earlier experimental neutron results

CONCLUSIONS AND PERSPECTIVES

Now

Most of samples are very complicated, involving-) many types of scatterers,-) many types of motions, no symmetry-) many time-scales

Models often offer a rather abstract form They can not easily be connected with experimental data.

Molecular dynamic simulations are very long

Pioneer time is finished

“Simple” samples, geometrically well described, stimulated: -) numerous scattering models especially for rotation -) original theoretical approaches based on symmetry

They evidenced the existence and the importance of EISF.

To go further

Collaborations are required:between experimentalists and theoreticiansbetween theoretical calculations and simulations

Multiply environment conditions:temperature, pressure, concentration, selective deuteration, illumination, …

Multiply experimental conditions:analyse sample with different instruments and/or resolutionsanalyse sample with different energy rangesR.E. Lechner Physica B 301 (2001) 83

use polarised neutrons to separate coherent from incoherent scatteringB. Gabrys et al. Physica B 301 (2001) 69

Improve software for data treatment:signal processing (intermediate scattering function)develop new analyses of data (memory function)

Collect maximum of information about the system:from other techniques from other scientific communities (chemistry, physics, biology,

biophysics, ….)