DATA ANALYSIS IN NEUTRON SCATTERING (AND BEYOND)

Miguel A. González(Institut Laue-Langevin, Grenoble, France)

D20

D11

Salsa

Figaro

Cyclops

IN8 IN6

IN16IN11

SOME EXAMPLES OF DATA ANALYSIS IN

NEUTRON SCATTERING

Outline

Neutron Scattering What do we measure?

Data reduction From neutron counts to S(Q) or S(Q,ω)

Background subtraction Absorption, attenuation, … Multiple scattering

Instrument simulations in data reduction

Data analysis Fitting, maximum entropy methods, … Computer simulations as a tool to analyze and understand expts

Software

We measure the number of scattered neutrons as a function of Q and ωAnd the result we are looking for is the scattering function S(Q, ω) that depends only on the properties of the sample

What do we measure?

∫∫ ω−⋅

π=ω ),(edd

21),( )( tGtS ti rrQ rQ

),(),('),2(dd

ds

2d

22

ω+ω=ωθωΩσ QQ SbSb

kkN

Probability density of having a given atom at (0, 0), and any atom at (r, t)

What do we measure?

bj depends on the nuclear spin state and the particular isotope!

– Coherent involves correlations between two nuclei (i, j)• Describes interference phenomena produced by the scattering of

neutron from all the nuclei in the sample.• Varies strongly with scattering angle

– Elastic Structure– Inelastic Collective modes

– Incoherent involves correlations between the position of an atom iat time 0 and the same atom at time t.

• No interference between waves scattered by different nuclei.• Isotropic scattering. • Single particle dynamics diffusive modes.

Coherent and incoherent scattering

Ideal instrument• No background• Uniform and collimated beam• Exact k0

• High angular definition• Detector efficiency = 1 @ all energies

An ideal experiment …

Ideal sample• Point-like (no multiple scattering) • High scattering section• No absorption• Fully coherent (or fully incoherent)

But nothing like this exists …One should try to prepare as well as possible the

experiment by optimizing the instrumental parameters and the sample quality.

It will depend on…The physical effect under investigation

The accuracy of the neutron experimental data

and available TIME(Almost) Unavoidable

measurementsEmpty container

Reference sample (Vanadium)

Unavoidable estimatesAttenuation

Multiple scattering

Various degrees of accuracyThe sample may allow employing somereasonable approximations:

Neglect of multiple scatteringT(θ) T(0)

Useful measurementsTransmissions

Absorbent sampleEmpty instrument

And a question …

A “full” analysis may last for ever and still be incomplete, so …

How much should we push the data analysis?

From neutron counts to S(Q)

Soller CollimatorsMonochromators

Detector (1536 cells, 0.1°)

variable takeoff

25x10 3

20

15

10

5

0In

tensit

y

140120100806040202Theta/°

Files: Hansen9:Applications:JFullProf:nacalf:210480Date of fit: 24/04/2003/ 10:34:13.0Na2Ca3Al2F14Chi2 = 5.65

D20 (ILL): A neutron diffractometer

From neutron counts to S(Q)

25x10 3

20

15

10

5

0

Inten

sity

140120100806040202Theta/°

Files: Hansen9:Applications:JFullProf:nacalf:210480Date of fit: 24/04/2003/ 10:34:13.0Na2Ca3Al2F14Chi2 = 5.65

I(2θ) I(Q)θ

λπ sin4

=Q

In some cases (almost) no treatment needed!

From neutron counts to S(Q)

D4 (ILL): A neutron diffractometer for liquids and glasses

From neutron counts to S(Q)

(Fischer, Rep. Prog. Phys. 2006)

Background subtraction needed!Absolute units ( S(Q) ) required: Normalization to an standard!

- Subtract scattering from container, environment (cryostat, furnace, etc.) and instrument.-Take into account attenuation of incident beam (due to absorption and scattering processes in the sample and the container).- Maybe necessary to measure also an absorbing sample (Cdor 10B) in order to separate sample-attenuated and non-attenuated backgrounds. - Correct multiple scattering effects.- Differential cross-section in absolute units normalize to vanadium (elastic incoherent scatterer) or other appropriate reference.- Static approximation (ħω << ħω0) inelasticity corrections.

Corrections needed

The background is sample dependent, so we need to perform two measurements:

Beam stop

Void sample position

1) An empty beam run (container without sample), IEB(θ,ω):Approximate measure of instrumental noise, but true background insample measurement will be less (because attenuation due tosample scattering and absorption).

2) A run with a “full” absorber (Cadmium or 10B4C) in place of thesample+container system, ICd(θ,ω): Measure electronic noise +other sources of noise not affected by the presence of sample +container.

Sample-like Cd specimenThe background to subtract to the sample

measurement can be estimated as:

- =Attenuated background

Ib(s+c) ≈ ICd + Tsc (IEB – ICd)

Instrumental effects: Background

( ) ( )xIxI Tµ−= exp0

When a neutron beam goes through a material, the beam intensity isattenuated because of the 2 possible processes able to remove neutronsfrom the beam, i.e. absorption and scattering:

0 L

dx

x

I0

µT is necessarily proportional to:-the ability of an atom to scatter and absorb (total cross section)-the number atoms (i.e. present removing units)

asTTT n σσσµ σ +== withThus,

Sample effects: Beam attenuation

( ) ( )[ ]LnI

LxIT 0T0

exp ωσ−==

=

The transmission of a slab sample is

Can we further generalize the “transmission” concept in order to take into account:• Any sample and beam shape• That some neutrons are scattered in the beam (i.e. forward) directionand that scattered neutrons too can be absorbed or scattered again

Overall attenuation will depend on the specific, θ-dependent, path afterscattering!A first improvement is the Paalman-Pings coefficient, first introduced for X-raysdiffraction (elastic scattering):

( ) ( ) ( )[ ]∫ −−= θµµθ 212 scatTincT ,PPexpdVV

A illill

so that ( ) ( ) ( ) ( )( )θθθ 222 11 IAI exp ≈0° 120°2θ

A(2θ)

0.68

0.70

Liquid in a cylindrical container

Sample effects: Beam attenuation

Due to absorption and multiple scattering in the sample we have( )( ) ( ) ( ) ( ) ( )( )θθθθ 2222 111

ss,sexp

ss IAII ≈≠

As,s(θ) is a “generalized transmission”(volume average of transmission over the possible paths contributing to the intensity at 2θ)

It depends on θ(because of sample geometry) It depends on ω

(because of the absorption dependence on E1)BUT INELASTIC EFFECTS ARE TYPICALLY NEGLECTED IN THE EVALUATION OF THE

ATTENUATION

Sample effects: Beam attenuation

1. It contributes to the measuredintensity (additional ‘background’). Thescattering from the container is (as forthe sample) attenuated,

2. It contributes to the attenuation ofthe signal from the sample.

3. In turn, the sample attenuates the scattering from the container. An empty canrun is NOT the true container contribution in a s+c measurement.

c,c c,sc c,c

s,sc

( ) ( ) ( ) ( )( ) ( ) ( )( )( ) ( ) ( ) ( )( )θθθ

θθθθθ

222

222221

,exp1

1,

1,

exp1

cccc

csccsscscs

IAI

IAIAI

=

+=+

Scattering by the container

(Paalman & Pings, J. Appl. Phys. 1962) (Petrillo & Sacchetti, Acta Cryst. A 1992)

Calculation of attenuation factors

Integrals evaluated either as a sum over finite elements on a grid or using Monte Carlo methods!

Calculation of attenuation factors

V(1.8 Å), Σs=Σa=0..36 cm−1

R=0.1cm

R=0.5cm

R=1cm

Rin=0.5 cm, Rout=0.6 cm

Rin=0.5 cm, Rout=1 cm

Rin=0.5 cm, Rout=1.5 cm

If a neutron is scattered once, it can be scattered again ...Multiple scattering affects the expected signal (i.e. the single-scattering intensity I(1),related to the double-differential cross section) in a two-fold way:

It removes singly scattered neutrons from the original direction.Therefore it ATTENUATES the single scattering intensitydetectable at a given angle

It contributes to the intensity detected at ‘another’ angle.Therefore it INTENSIFIES the signal, mixing up with the truesingle scattering component at a given angle.

Multiple scattering causes both the loss of “good” neutrons and the detection of “bad” ones!

( )+≈+≈ exp)2(exp)(exp)(exp1exp with IIIII mm

Multiple Scattering

Big problem: The evaluation of MS requires to know exactly what the neutron does in the sample, i.e. S(Q,ω), which is not known!

A priori experimental “correction”: Try to MINIMIZE it!

Level 0: reduce sample size (basically thickness) in order to have a large transmission (rule of thumb: T > 90%)

Multiple Scattering

Level 1: subdivide the sample in a series of smaller samples by using absorbing spacers parallel to the incident beam.

A posteriori correction: Compute MS and remove it using approximate models for S(Q,ω). ITERATIVE PROCESS!

Neglect MS processes higher than second-order scattering and approximate S(Q,ω) as ELASTIC AND INCOHERENT.

(Vineyard, Phys. Rev. 1954)

Estimating MS using simple approximations

(Blech & Averbach, Phys. Rev. 1965)

Slab geometryRatio of 2nd/1st order scattering vs σTtσTt < 0.05 2nd/1st < 0.1E.g. Vanadium (6 Å): σT = 1.55 cm−1 t ≈ 0.3 mm (T ≈ 0.95)σS = 0.36 cm−1 t ≈ 1.4 mm (T ≈ 0.80)

Refining the estimation of MS

(Soper & Egelstaff, NIM1980)(Sears, Adv. Phys. 1975)

Beam width = 1 cm

Beam width = 0.5 cm

Beam width = 0.25 cm

--- Uniform profile Gaussian profile But still assuming elastic & isotropic scattering!

And beyond the isotropic approximation

Effective scattering function = Singly scattered neutrons + Container + Multiple

Probability that an incident neutron with k0 is scattered

at r into k:

Distribution of neutrons after 1st scattering(t(E0) = sample transmission)

Fraction of singly scattered neutrons(scattering power of the sample)

(Sears, Adv. Phys. 1975; Palomino et al., NIMB 2007)

But these are not the neutrons detected!

And beyond the isotropic approximation

Distribution of neutrons detected after 1st scattering:

H1(k0,k) = Attenuation factor

In diffraction experiments:

And beyond the isotropic approximation

Incoherent model:

Coherent model:

Multiple scattering factor:

Generalized attenuation factor = Ratio of detected singly scattered neutrons to total singly scattered

neutrons

And beyond the isotropic approximation: Monte Carlo simulations

INPUT DATA REQUIRED:• Total cross section of sample and container as a function of energy• Absorption cross section of sample and container as a function of energy• Detector bank efficiency as a function of energy• Geometry parameters of the instrumental setup and sample environment•A model describing the scattering law of the sample and container

And beyond the isotropic approximation: Monte Carlo simulations – H2O

(Palomino, NIMB 2007)

And beyond the isotropic approximation: Monte Carlo simulations – D2O

(Palomino, NIMB 2007)

And beyond the isotropic approximation

(Palomino, NIMB 2007)

•To normalize the data we need to determine the experimental factorF = Φ η(ω0) N ∆Ω.

•Most of these quantities are known only approximately, in particularthe incident neutron flux at the sample position.

•Thus a specific measurement is required, using a REFERENCE sample ofwell-known scattering properties.

•A solid is usually the choice because of its mostly elastic scattering,though this condition is not mandatory: it is only important that thedifferential cross-section of the reference sample is a known quantity.

•An incoherent scatterer is the best choice, because a non-dramaticchange in intensity with varying Q (Bragg peaks) is required fornormalization purposes (flat diffraction pattern).

In most neutron experiments the reference is VANADIUM (but also water in SANS experiments)

Normalization to absolute units

Differential cross section

(Palomino, NIMB 2007)

The double-differential cross section for nuclear neutron scattering is

( ) ( )∫ ∑+∞

∞−

⋅⋅−∗−=Ω βα

βαω βα

πωσ

,

tiiti eebbN

edtkk

ddd RQRQ 0

0

12 1

21

From the theory of space- and time-dependent correlation functions:

( ) ( ) ( )∑∫=

⋅⋅−∞+

∞−

−=N

,

tiiti eeN

edt,QS1

0121

βα

ω βα

πω RQRQ

From dσ/dΩ to S(Q)

( ) ∫+∞

∞−= ),( ωω QSdQSand

( ) ( )

( ) ( ) !!QSBA,QS~kkkdN

dddkdN,IdI

tcos

tcostcos

+=

=≈=

≠≠∫

∫∫

=∞−

=∞−

=∞−

ωω

ωωθωθ

ω

θ

ω

θ

ω

θ

0

11

2

2

1

22

)η( ΔΩΦ

ωΩσ)η( ΔΩΦ22

0

00

But we measure:

If E0 >> ħω :

elQkkQ =→−−

−+= θ

ωωθ

ωω sin212 cos211 0

000

20

0

20

21 1 kkk →

−=ωω

∫∫∞+

=∞−∞−

elQtetancosQtetancos

~0

2

ω

θ

( ) ( ) 00101 constant Efixedatkkkk ==⇒= ηη

( ) ( )

( ) 2coh

2inc0

2

cost

0

cost

0

2

1

2

andwithΩσ)η( ΔΩΦ

ωΩσ)η( ΔΩΦ

)η( ΔΩΦωΩσ)η( ΔΩΦ2

0

0

bBbA!!QSBAddkN

ddddkN

,QS~dkNdd

dkdNI

Qtcos

∝∝+=

==

→≈

=

∫

∫∫

=∞−

+∞

=∞−

=∞−

ω

θ

ω

θ

ω

ωωωθ

Static approximation

and

Inelasticity effects

P(q) = polynomial expansion in powers of q2 and (mn/Mα)

(Fischer, Rep. Prog. Phys. 2006)

From neutron counts to S(Q)

D11 (ILL): A small angle neutron scattering difractometer

Empty beam Blocked beam Water

Measurements to do in a SANS exptSample

+ Transmissions of sample and empty cell

CdEC

sS

S

sSSS I

dd

VCT

dd

VCTdI +

Ω+

Ω=

σλσλθ 1)(1)()(

CdEC

sECEC I

dd

VCTI +

Ω=

σλθ 1)()( )()()()( θλελφλ ∆Ω= AC

)()()(

)()()()(

)(1 1−

∆Ω

−−

−

=

Ωcm

dT

IITT

II

dd

V S

EC

CdEC

ECS

CdS

S

s

φθεθ

θθθθ

θσ

)()( θεθ →WI

Absolute differential cross section

- Dead time corrections(Brûlet, J. Appl. Cryst. 2007)

Needed corrections

-Solid angle of each detector cell

τnnnm +

=1

- Angle dependent transmission

0

1cos

1

0

lncos

11

1)(T

TTT

−

−=

−

θ

θθ

- Detector efficiency with θ and λ (instrument dependent)

θθ 3cos)0()( ∆Ω=∆Ω

D11 data

λ=10 Å, d=4m, coll=4m

λ=10 Å, d=1.2 m, coll=4 m Corrected 2D data

From neutron counts to S(Q,ω)

(Bustinduy, Rev. Sci. Inst. 2007)

Modern TOF spectrometers

- Need to manipulate and visualize very large files.E.g.: IN5 = 384 PSD tubes x 256 pixels/tube x 1024 tof channels ≈ 108 (768 MB)

- Visualize 4D datasets: I(Qh,Qk,Ql,ω)

- Rebin data in an optimal way(adaptative binning algorithms)

Equal width binning Multiresolution algorithm Adaptative tessellation

From neutron counts to S(Q,ω)

(Copley, J. Res. NIST 1993)

Interpolate from (2θ,ω) to (Q,ω)

)cos()(222 000

22

θωω

−−−= EEEmQ

From neutron counts to S(Q,ω)

(Farhi, 2011)

Data treatment in the next futureVirtual experiments using McStas

n

TOF spectrometer

D2

last scattering ?

D1

scattering point

scattering direction ks

next component

monochromator

sample

McStas (instrument simulation) + S(Q,ω) from experiment, classical or ab initio simulations.

Virtual experiments using McStas

The only way to fully include all the instrumental (background & abs/scat by container and sample environment) + sample (abs/MS) effects!

Virtual experiments: l-Rb in IN5

Total signal

Container (Nb)

Coherent scattering (Rb)

Incoherent scattering (Rb)

Multiple scattering x 100

Cryofurnace (Al) x 10

Analyzing S(Q) or S(Q,ω)

A powder diffractogram

Use Rietveld refinement

The Rietveld Method consists in refining a crystal (and/or magnetic) structure by minimizing the weighted squared difference between the observed and the calculated pattern against the parameter vector: β

22

1( )

n

i i cii

w y yχ β=

= −∑ 21i

iwσ

=

The Rietveld method

• Non linear problem Iterative procedure• The least squares procedure is a local optimization method. • It provides (when it converges) the value of the parameters constituting the local minimum closest to the starting point• A set of good starting values for all parameters is needed• If the initial model is bad for some reasons the LSQ procedure will not converge, it may diverge.• Several quality factors usually employed:

The Rietveld method

∑∑ −

=i

ciip y

yyR

( )( )

2/1

2

2

−=

∑∑

ii

ciiiwp yw

yywR

)'(')()'(' )()(

obsIcalcIobsI

Ri

hklhklB ∑∑ −

=

( )2/1

2

−=∑ ii

e ywPNR

Rietveld method: SrC3O7H3

Acid strontium oxalate from XR

Disordered systems

D2O

HDO

H2O

(Soper, J. Chem. Phys. 1997)

∑ −=

k

kE

koC

oQSQS

2

2,2 )()(

σχ

Disordered systems: Reverse MC1. Initial configuration, with PBC2. Calculate partial p.c.f.’s & S(Q)

j

oC,joC

ij rr4

(r)nrg

ρπ ∆= 2

, )(

( )∑∑ −=i j

ijjijioC rgbbccrG 1)()(,

∫∞

=−0

,, sin)(41)( QrdrrrGQ

QS oCoC πρ

3. Calc. difference with expt

4. Move one atom at random5. Recalc. S(Q) and χ2 for new position6. If χ2

old > χ2new accept move. If not

accept move with the probability

( ) 2/exp 22on χχ −−

• In general, RMC produces the most disordered structure consistent with the experimental data and constraints applied.• Non-uniqueness: More than one model compatible with the data can be produced. Good to add information (n + XRD, density, EXAFS, …) • Requires accurate, correct data.• Constraints required to treat molecular or covalent systems.

Disordered systems: Reverse MC

0 2 4 6 8 10-1.0

-0.5

0.0

0.5

S(Q

)-1

Q(Å-1)

0 2 4 6 8 10

-0.5

0.0

0.5

1.0

S(Q

)-1

Q(Å-1)

H2O, X-rays (Narten&Levy, 1971) D2O, neutrons (Soper et al., 1997)

Structure of liquid water

Empirical Potential Structure Refinement

QENS data

Hydrated Nafion membrane (Perrin, J. Phys. Chem. C 2007)

( )[ ]

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

1),(

)1()( )(),(

0

22

2

ωωωδω

ωπω

ωδωω

∆+=

+=

⊗−+⊗=

∑ QLQAQAQS

QDQDQS

SSAARQS

lR

t

tT

RTfixfix

meas

Bayesian analysis of QENS data

(From L.C. Pardo, UPC, Barcelona)

Bayesian analysis of QENS data

(Sivia, Physica B 1992)QENS data relatively poor to distinguish between 2 or more quasielastic components!

Model data generated using one δ and two lorentzians of widths 50 and 250 µeV

Bayesian analysis of QENS dataThe two large lorentzians correspond to two types of internal motions, but the central line is a δ or a broadened line?δ Long-range motion invisibleBroadened LR motion observable

QENS spectrum of DMPC at 30°C(Busch, JACS 2010)

Data agree with flow-like motion!

Computer simulations in data analysis

Replace experiments whenever they are impossible or dangerous Astrophysics, nuclear accidents, earthquakes, ... High temperatures and pressures No stable phases (e.g. water in “no man’s land”)

If a good model is available, they are a cheap and easy way of obtaining reliable data

Rapid development of hardware and software Minimize and optimize expensive experiments

Provide unique information Behavior of particular atoms Compute special correlation functions Test theories (normally based on ideal models)

Computer simulations in data analysis

Neutrons see nuclei Van Hove correlation functions: S(Q,ω) ↔ F(Q,t) Much more direct link between simulations and neutron scattering that with other techniques (specially for classical MD)

Same (Q,ω) range time: fs to ns (up to µs in some cases;

only some ps with ab initio DFT) length: Å to several nm (few Å with ab initio DFT)

Increasing complexity of systems investigated Analytical models insufficient to interpret the experiment Simulations allow to understand better the data

DFT and phonon calculations

C

5

C

5

C

5

C

5

R2

R1

CCC C

4

C5

C N1

N

xz

yO

1

R2

0.4 0.8 1.2 1.6 2.0 2.4 2.80.25

0.50

0.75

1.00

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

A 0(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

babi

lity

(b) Crystal State (c) Liquid State

(c)

(a)

(b)

0.5 A

(d)

R1

R2

2

3

1

4

MD and complex motions

1E-5

1E-4

1E-3

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.01E-8

1E-7

1E-6

1E-5

1E-4

1E-3

373K

(b) Liquid

300K

S(Q

,E) (

meV

-1)

Resolution

Q=0.6 Å-1

Q=1.0 Å-1

Q=1.6 Å-1

E (meV)

(a) solid

(Aoun, J. Phys. Chem. Letters 2010)

Software

LAMP: Large Array Manipulation Program

• ILL, since 1995• Based on IDL (more recent version 8.1)• Initially based in “user macros” flexibility• Recently added “spreadsheets” for automatic treatments

LAMP: Large Array Manipulation Program

DAVE: Data Analysis and Visualization Environment

• NIST, since 2002• Based on IDL• Large library of modules and tools

MANTID: Manipulation and Analysis for ISIS Data

• ISIS, project started in 2007• 2010: Collaboration with HFIR and SNS (USA)• C++ & Python• Create a data analysis framework easily extendable

Conclusions

• Data corrections will depend on the kind of instrument, as well as on the sample and the physics we are interested in.

• Generally, we will need to subtract the background (taking into account absorption and self-shielding), normalize the data to a known standard, and perhaps correct for multiple scattering.

• A complete correction requires an iterative process including a very detailed simulation of the full experiment (instrument + sample + sample environment).

• Almost never done (impossible up to know, so rely on reasonable approximations), but computational tools are now ready (e.g. McStas) and this could become the standard way of data analysis in the future.

• Computer simulations and modeling are essential to understand complex data and/or systems.

Thank you!