Diffrazione ad “alti” angoli su un 2 Diffrazione ad “alti” angoli su un 2 assi:assi:

i problemi, le misure & l’analisi datii problemi, le misure & l’analisi datiEleonora GUARINI Eleonora GUARINI

Dipartimento di Fisica, Università di FirenzeDipartimento di Fisica, Università di Firenzeguarini@fi.infn.itguarini@fi.infn.it

Giornate Didattiche 2010 Giornate Didattiche 2010

Hotel Steinpent, S. Giovanni in Valle Aurina 27 – 30 Giugno 2010

Società Italiana di Spettroscopia Neutronica

OutlineOutline

What we measure Versus what we look for

The ideal neutron scattering experiment

A first step towards the real case: neutrons in a material

The real experiment: effects influencing the neutron measurements

Basic treatment of neutron diffraction data

Tailoring and Performing an experiment on a 2-axis diffractometer

The sought-for quantityThe sought-for quantity

The central quantity in the study of the microscopic structure of an isotropically scattering sample (a liquid, a powder…) is

N

,

ii eeN

,QSdQS1

001

RQRQ

STATIC STRUCTURE FACTOR

The accessible quantity The accessible quantity On a two-axis diffractormeter (fixed incident energy, angular dispersive) we measure:

2I

Origin of the accessible Origin of the accessible quantity Iquantity IThe double-differential cross section for nuclear neutron scattering

(b’s are the scattering lengths) is

,

tiiti eebbN

edtk

k

dd

d RQRQ 0

0

12 1

2

1

the probe the probe fingerprintfingerprint

While, from the theory of space- and time-dependent correlation functions, it is found:

N

,

tiiti eeN

edt,QS1

01

2

1

RQRQ

Let’s make the resemblance between double-differential cross section and S(Q,) more evident as:

,

tiiti eebbN

edt,QS~

,QS~

k

k

dd

d

RQRQ 0

0

12

1

2

1

Were the scattering lengths (i.e. the fingerprint of the probe we are using) absent, this would be exactly the dynamic structure factor

with

ωΩ

σ)η( ΔΩΦ2

2

1 dd

dNk,I

beam detector sample

Origins II… (in the ideal Origins II… (in the ideal casecase))

2k0

k1

2Q

k0

k1 2

120

2

10

10

10

2

,,

kkEEE

kkfQ

m

kkQ

Restrictions:no magnetic effects, no polarizations.

Assumption:fixed incident energy E0

Energy analysis of emerging neutrons

(,)

In the ideal case ALL relevant quantities are exactly definedA “size-less” perfectly straight beam (of flux ) is supposed to impinge on a “size-less” sample composed of N atoms, all equally exposed to the beam (a paradox, nearly!).

Even in such ideal conditions, the “measured” signal

would NOT coincide with the double-differential cross section. The latter, in turn, DOES NOT give immediately the dynamic structure

factor

Counts per unit time and unit

frequency interval

Origins III… Origins III…

Our 2-axis diffractometer is UNABLE, however, to perform an energy analysis

of the emerging neutrons…What we actually collect at the detector is:

!!QSBA,QS~

k

kkdN

dd

dkdN,IdI

tcos

tcostcos

0

11

2

2

1

22

)η( ΔΩΦ

ωΩ

σ)η( ΔΩΦ22

0

00

This quantity differs EVEN from the differential cross-section, which is defined as:

related to related to the probe the probe fingerprintfingerprint

ωΩ

σ2

2

2

0

dd

dd

d

d

tcos

because of the detector’s efficiency dependence on the energy of the scattered neutrons.

BUT,BUT,if inelasticity effects are if inelasticity effects are

SMALL…. SMALL…. i.e., … if E0 >> E , then:

elQkkQ

sin212 cos211 000

0

20

0

20

21 1 kkk

elQtetancosQtetancos

~0

2

00101 constant Efixedatkkkk

and also:

2coh

2inc0

2

cost

0

cost

0

2

1

2

andwithΩ

σ)η( ΔΩΦ

ωΩ

σ)η( ΔΩΦ

)η( ΔΩΦωΩ

σ)η( ΔΩΦ2

0

0

bBbA!!QSBAd

dkN

dd

ddkN

,QS~

dkNdd

dkdNI

Qtcos

per un monoatomico e monoisotopico

Dreaming a little Dreaming a little more...more...

The idealized diffraction experimentidealized diffraction experiment would require the combination of:

- a “perfect” instrument- an “ideal” sample

- a direct I(2) S(Q) relation

the perfect the perfect diffractometerdiffractometer

the ideal samplethe ideal sample

No background

Uniform and collimated beam

Perfect k0 definition

High angular resolution

Detector efficiency 100%

Some features ofSome features of

Bare and highly scattering

No absorption

“point-like” (no size, no MS)

Fully Coherent

Simple

“Extremely interesting”

TO OPTIMIZE YOUR EXPERIMENT AND SAMPLE PREPARATION YOU NEED TO KNOW WHAT ARE the PROBLEMS IN NEUTRON SCATTERING AND DIFFRACTION…

xIxI T exp0

Towards the real Towards the real case .I.case .I.Beam attenuationBeam attenuation

When a neutron beam goes through a material, the beam intensity is attenuated because of the 2 possible processes able to remove neutrons from the beam, i.e. absorptionabsorption and scatteringscattering.

Consider the simplest case where a uniform, collimated, and monochromatic beam crosses a homogeneous slab sampleslab sample perpendicular to the beam axis:

0 L

dx

x

I0

What is What is TT ? ?T 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 withThus,

Towards the real case .II.Towards the real case .II.Transmission & Attenuation Transmission & Attenuation coefficientscoefficients

LnI

LxIT 0T

0

exp

Fraction of

transmitted neutrons

The transmission of a slab sample is

Of course, R = 1 -T is the fraction of removed (scattered or absorbed) neutrons.

Can we further generalize the “transmission” concept in order to take into account:-any sample and beam shape- that some neutrons are NOT scattered in the beam (i.e. forward) direction and that scattered neutrons too can be absorbed or scattered again (the overall attenuation should depend also on the specific, -dependent, path after scattering)

?A first improvement is the Paalman&Pings coefficient,first introduced for X-rays diffraction (elastic scattering):

y

0x

2P 2

12 scatTincT ,PPexpdV

VA ill

ill

so that 222 11 IAI exp

0° 120°2

A(2)

0.68

0.70

Liquid in a cylindrical container

Towards the real case .III.Towards the real case .III.Multiple scattering Multiple scattering

If a neutron is scattered once, it can be scattered If a neutron is scattered once, it can be scattered twice... (thrice, etc.)!twice... (thrice, etc.)!

Multiple scattering affects the sought-for 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 single scattering intensityintensity detectable at a given angle

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

Multiple scattering causes both the loss of Multiple scattering causes both the loss of “good” neutrons“good” neutrons

and the detection of “bad” ones! and the detection of “bad” ones! Generally, one can write:

exp)2(exp)(exp)(exp1expwith IIIII mm

The multiple scattering intensity can be evaluated, starting from the double contribution, by Monte Carlo integration. Different degrees of approximation are possible.

2

k1 + k1k0 + k0

2e

The real-life situationThe real-life situation R

eal in

str

um

en

tR

eal in

str

um

en

tR

eal sam

ple

Real sam

ple

Effect Uncertainty on Main consequence Background noise (hopefully constant in time!) Zero-signal level

Beam divergence Focussing devices 2, E0, paths in the sample

Monochromator characteristics: crystal quality

E0

Finite detector size 2, paths in the sample

Affect Q and E resolution

Detector efficiency Signal attenuation and distorsion

Container scattering and absorption Bare sample signal “Background”, attenuation

Sample absorption Signal attenuation

Sample scattering Multiple scattering

Finite sample size 2, paths in the sample

Q resolution,

absorption multiple scattering

Incoherent too and “complex” ……. ……..

,MS

Instrumental Instrumental effects .I.effects .I.

Background noiseBackground noiseIt partly depends on the sample+container, and requires specific measurements. In principle, one should perform: Beam stop

Void sample position

1) An empty beam run (no sample+container) in order to collect IEB(,) which approximates the thorough instrumental noise.2) A run with a “full” absorber (typically Cadmium) in place of the sample+container system. The collected ICd(,) is due to those background neutrons which are unaffected by the presence of sample + container.

Sample-like Cd specimen

That part of background which is modified by the sample+container can be estimated as:

- =Attenuated background

I b(s+c) ICd + Tsc (I EB – I Cd) (changing with the sample density)

Instrumental Instrumental effects .II.effects .II.

Detector efficiencyDetector efficiencyIt is measured by the absorbing power of the detection system, i.e. by

the coefficient R seen before, with T = abs. Efficiency depends on:a) the specific absorbing material (abs) b) geometryc) energy of the neutrons (scattered by the sample and) reaching the detector, because of the absorption dependence on energy.

For a slab detector with gas density nD and thickness L:

Lknk D 1abs1 exp1

For a cylindrical detector perpendicular to the beam, with radius r and density nD:

r

y

0

dy

x

L(y)

r

rkn

yrkndyr

k

D

r

D

1abs

0

221abs

1

2exp1

2exp1

1

Sample-related Sample-related effects .I.effects .I.AttenuationAttenuation

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

ss,sexp

ss IAII

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

It depends on (because of sample geometry)

It depends on (because of the absorption

dependence on E1)BUT INELASTIC EFFECTS ARE TYPICALLY NEGLECTED IN THE

EVALUATION OF THE ATTENUATIONThe expression of As,s (2) can however be derived by taking geometrical and size effects into account...

Attenuation & size-effectsAttenuation & size-effects

Let’s write our best expression for Is(1) exp(2):

D,P,

dd

dd

D,PL

cos

PdSdVnI

scatTtrue

incT

S

d

V

ills

exps

dill

s

1

2

12

01

exp2

exp2

0

with = 0 - 1 AND L = L(P,D) = (P,D) true = true(P,D)

If we neglect the (weak) dependence of 2true on P and D, as well as the energy dependence of T over the scattered flight path, we can still factor out a sort of “Paalman&PingsPaalman&Pings" coefficient including size-effects:

2

0022

11

exp

220

D,PL

cosdSdV

D,PPD,PL

cosdSdV

,dd

ddNI

dill

s

dill

s

S

d

V

ills

scatTincT

S

d

V

ills

exps

Sd /LD

2

222 11s,ss

exps AII

θtrue

L

D

θ

LD

P

dSd

0

1

Correction for Correction for AttenuationAttenuation

A priori experimental “correction”A priori experimental “correction” (minimization):Level 0: use of low absorption samples if possible (isotopic substitution)

Level 1: use of thin and symmetry-adapted samples, compatibly with intensity needsLevel 2: ‘maximize’ the incident energy, compatibly with the property under study(your wavelength should fit your “d-spacings”!!)

A posteriori correctionA posteriori correction:

Expt.orCalc.

For each 2 of the experiment, calculate As,s()!

The only practicable way is by Monte Carlo integration. Still, it is a time-consuming procedure.(Otherwise, simulate directly the single-scattering intensity… but youneed a good model for the unknown double differential cross section…)

To simplify things, it is often assumed that:

As,s(2) As,s( = 0, = 0)= Ts

Complication: Complication: the the containercontainerIn neutron experiments on liquids the container plays an important

role. It contributes to the measured intensity (additional ‘background’). The scattering from the container is (as for the sample) attenuated:

1.

It contributes to the attenuation of the signal from the sample.

2.

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

3.

c,c c,sc c,c

s,sc

expcs

expcs

exps

cc,cexp

c

csc,cssc,sexp

cs

ITII

IAI

IAIAI

111

11

111

222

22222

Correction for Correction for attenuationattenuation

in the presence of a in the presence of a containercontainer

A priori experimental “correction”A priori experimental “correction” (minimization):

A posteriori correctionA posteriori correction:

Level 0: use of low absorbing and low scattering materials

Level 1:

Level 2: ‘maximize’ the incident energy, compatibly with resolution requirements

use of thin containers, compatibly with sample conditions (e.g. pressure) or environment (cryostat/furnace).

For each 2 of your experiment, calculate Ac,c(2), Ac,sc(2) and As,sc(2) !!!

Example:

dill

c

dill

c

SV

dill

c

SV

dill

c

sccdSdV

adSdV

A

exp

,,

D,PPD,PPa sscat

sT

sinc

sT

cscat

cT

cinc

cT 0000

Monte Carlo integration(P,D) couples are randomly sampled over the 5-dimensional space (Vc

ill + Sd)

(in the elastic approximation)

PAUSA?????

Sample-related Sample-related effects .II.effects .II.

Multiple scatteringMultiple scatteringDue to multiple scattering in the sample we have

222

222232

11

exps

exps

expms

expms

exps

exps

exps

III

IIII

But we can’t measure Is(m)exp and we don’t know the scattering law.

L

θθ1

θ2

LD

D

P1

dSd

P2

1 2

0

Correction for Multiple Correction for Multiple ScatteringScattering

A priori experimental “correction”A priori experimental “correction” (minimization):

A posteriori correctionA posteriori correction:

Level 0: use of “low”-scattering, “small” and beam-adapted samples, compatibly

Level 1: subdivide the sample in a series of smaller samples by using absorbing

with intensity requirements

spacers parallel to the beam direction.

It is based on the use of approximate models for (Models can be refined by an iterative procedure)

Given a model, a way is to simulatesimulate multiple scattering processes

dd

d

2

follow many neutron histories with a cutoff weight depending on attenuationconstruct a distribution function at the detector

Alternatively, calculatecalculateby Monte Carlo

integration

22 32 exps

exps I,I

(P1,P2,D) triplets are randomly sampled over the 8-dimensional space (Vs

ill + Vs + Sd). (in the elastic approximation)

(P1,P2,D) triplets are randomly sampled over the 8-dimensional space (Vs

ill + Vs + Sd). (in the elastic approximation)

1

12 1

0

1

1

11

2

1 sn

ns

n

ns

ms

s

sn

s

ns IIIIn,

I

I

I

I

A well known approximation (good for slab samples) is often used to avoid the calculation of I(3)…I(n):

A well known approximation (good for slab samples) is often used to avoid the calculation of I(3)…I(n):

Vineyard approximationVineyard approximation

Complication: Complication:

multiple scattering involving multiple scattering involving the containerthe container

cc,c cc,sc cc,c

As for the single scattering, the true multiple scattering contribution from the container in a s+c run is not that present in the empty cell measurement (attenuation due to the sample).

Moreover “cross” multiple scattering events can occur, involving both sample and container.

sc,sc cs,sc

222

2221

1

expmc

expc

expc

expmcs

expcs

expcs

IIBI

IIBI

Multiple scattering correction in a multi-Multiple scattering correction in a multi-element systemelement systemsample + container (+ cryostat/furnace + ...)sample + container (+ cryostat/furnace + ...)

While single scattering intensities are easily generalized and accounted for in the case of a multi-component system (it “only” requires the appropriate empty-“element” measurements and the calculation of the consequent

absorption coefficients), the same is not true for multiple scattering.

Complication arises from the cross contributions [which may not be negligible for some (forced) combinations of the sample and the container macroscopic scattering properties].

In the simplest case (s+c) we have (for each 2)

mc,cc

mc

msc,cc

msc,cs

msc,sc

msc,ss

mcs

II

IIIII

But we skip the details…..

Handling experimental quantities and Handling experimental quantities and calculations calculations

We have:

eb

Cd

c

cs

I

I

I

Iexp

exp

CdebcCd

CdebscCd

mcc

mcscs

IITIB

IITIB

IIB

IIB

expexp1

expexp1

exp2,

exp1

exp2,

exp2,

exp2,

exp2,

exp1exp1

cccc

scscsccsscccscsscs

IIB

IIIIIIB

EXP

calc

cccc

calcsccsccsscs

IAB

IAIAB

1

11

,

1,

1,

exp

exp

c

cs

I

I

with

calc

cc

calc

cccc

calc

sccscs

calc

scscsccsscccscsssc

I

I

II

IIII

1,

2,

1,

1,

2,

2,

2,

2,

The single-scattering The single-scattering intensity intensity IIss(1)(1)

sc,s

CdebcCdexp

ccalccc,c

sc,ccalcsc

CdebscCdexpsc

s A

IITIIA

AIITII

I

111

22

1

inccoh

ELASTIC

s

bQSbF

QS~

Fd

dFI

ELASTIC

If multiple scattering is neglected

If attenuation coefficients are approximated by the transmissions(As,sc ≈ Ac,sc ≈Tsc ≈Ts Tc and Ac,c≈Tc)

sc

Cdexp

csCdexpsc

s T

IITIII

1

F is the instrumental factor due to flux, solid angle, detector efficiency and sample density … we need to MEASURE it

VanadiumVanadiumVanadiumVanadium

Data normalization to Data normalization to absolute unitsabsolute units

To normalize the data we need to determine the experimental factor F = (0) N .

Most of these quantities are only approximately known, the flux at the sample mainly…( it depends on too many variables: source spectrum, monochromator, collimators… andtheir REAL performances).

Thus a specific measurement is required, using a REFERENCE sample of well-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 the differential cross-section of the reference sample is a known quantity (e.g. gaseous hydrogen).

An incoherent scatterer is the best choice, because a non-dramatic change in intensity with varying Q (Bragg peaks) is required for normalization purposes (flat diffraction pattern).

In MOST (not ALL) neutron experiments the reference sample is

To normalize the data we need to determine the experimental factor F = (0) N .

Most of these quantities are only approximately known, the flux at the sample mainly…( it depends on too many variables: source spectrum, monochromator, collimators… andtheir REAL performances).

Thus a specific measurement is required, using a REFERENCE sample of well-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 the differential cross-section of the reference sample is a known quantity (e.g. gaseous hydrogen).

An incoherent scatterer is the best choice, because a non-dramatic change in intensity with varying Q (Bragg peaks) is required for normalization purposes (flat diffraction pattern).

In MOST (not ALL) neutron experiments the reference sample is

Data normalization to Data normalization to absolute unitsabsolute unitsWhat we need is only a proportionality relation (… neglecting inelasticity

effects). In general, measurements on the sample and on the calibration sample (e.g. vanadium) may be synthetized as:

where different GEOMETRIES of the sample specimen and of the calibration sample are taken roughly into account through an overall “solid-angle” effect.

If we try to make the geometrical differences (between sample and reference) as small as possible, then REF SAMPLE (common) and therefore:

What we need is only a proportionality relation (… neglecting inelasticity effects). In general, measurements on the sample and on the calibration sample (e.g. vanadium) may be synthetized as:

where different GEOMETRIES of the sample specimen and of the calibration sample are taken roughly into account through an overall “solid-angle” effect.

If we try to make the geometrical differences (between sample and reference) as small as possible, then REF SAMPLE (common) and therefore:

)UNKNOWN(SAMPLE

SAMPLE)ILLUM(

SAMPLEs

)KNOWN(REFREF

)ILLUM(REFREF

d

dNI

d

dNI

01

0

2

2

)UNKNOWN(SAMPLE

)KNOWN(REF)ILLUM(REF

REF)ILLUM(

SAMPLE

s

)KNOWN(REF)ILLUM(REF

REF

d

d

dd

N

INI

dd

N

I

22

2

1

0

The data-analyzer endless The data-analyzer endless enigmaenigmaHow much should one push the refinement of the data

analysis and related calculations?

an “extremely accurate” analysiswould still be approximate and would never

end!It depends on…The physical effect

under investigationThe accuracy of the neutron

experimental data

and available TIME

Unavoidable measurements

BackgroundEmpty container

Reference sample (Vanadium)

Unavoidable estimatesattenuation

multiple scatteringVarious degrees of accuracythe sample features may allow for reasonable approximations down to:

use of transmissionsneglect of multiple scattering

Useful measurements

Transmissions

Riassumiamo un po’…Riassumiamo un po’…

La quantità fisica importante in esperimenti di diffrazione è S(Q), ma ciò che si misura non è connesso in modo diretto ad essa.L’intensità direttamente misurata soffre di vari effetti come il background, attenuazione, scattering multiplo, efficienza e, ovviamente, della risoluzione FINITA dello strumento. Molti di questi richiedono misure specifiche.

Anche in un caso ideale e ammettendo di liberarsi facilmente da fattori strumentali, la sezione d’urto differenziale NON è ciò che si cerca perché contiene ancora le sezioni d’urto e perché lo scattering NON E’ in generale ELASTICO.

In un esperimento reale, vanno ottimizzati i contenitori, il campione stesso, e predisposte varie misure, sia principali (campione nel contenitore e contenitore vuoto) che ancillari (vanadio, cadmio, trasmissioni). Ma dipende anche dal sample…

Altri step sono obbligatori in generale: e.g. normalizzazione delle intensità misurate nei vari run agli stessi conteggi di MONITOR

(cosa è fondamentale ricordare indipendentemente dallo specifico strumento)

Facciamo finta di arrivare Facciamo finta di arrivare sul D1A di ILL…sul D1A di ILL…

… e di dover fare delle misure su un liquido.

A prioriavrete scelto un contenitore adatto al vostro liquido in termini di:

Potere di scattering del campione corrispondente…

Ps S / T (1-T) con T dipendente da n e dallo spessore di campione illuminato (cambia con la forma del contenitore e del fascio…)

Il potere di scattering vi dice anche quanto multiplo vi potete aspettare. In genere si cerca di non superare il 20%....

Proprietà di scattering del materiale della cella (min. scattering, min. assorbimento, possibilmente INCOERENTE)

Compatibilità di materiali fra cella e campione (ci sono campioni corrosivi…)

Facciamo finta di arrivare Facciamo finta di arrivare sul D1A di ILL…sul D1A di ILL…

A posteriori (in genere, ma dipende…):

Preparate la cella(schermi di Cadmio, se necessario)

Adattate con diaframmi la dimensione del fascio alla parte utile della cella

Posizionate al meglio la cella sul fascio(se è slab va cercata la posizione ortogonale al fascio)

Verificate la posizione coprendo completamente la celladi Cadmio (prendete vari riferimenti geometrici per poterla riposizionare bene, ma solo con Cd esterno)

Facciamo finta di …Facciamo finta di …A posteriori (in genere, ma dipende…):

Va scelta l’energia incidente di lavoro e la collimazione(cercando il solito compromesso fra intensità, assorbimento e risoluzione…: ma dipende dal campione e dalla proprietà cercata!!)

Vanno scelti i tempi di misura di campione+cella, cella vuota, assorbitore, campione di riferimento… in base a:tempo assegnato sullo strumentoaccuratezza che si desidera raggiungere sull’intensità dal solo sample (cioè dopo la sottrazione della cella)potere e proprietà di scattering di cella e campione

Vanno eseguiti vari sottorun per ciascun gruppo di misure (in modo da controllare la stabilità del campione e dello strumento)

I sottorun compatibili (normalizzati allo stesso conteggio di monitor e in accordo fra loro) vanno raggruppati tramite una media pesata: questi saranno i file RAW necessari per la successiva analisi. Ne avremo uno per tipo: sample+cella, cella, assorbitore, campione di riferimento etc….

I file raw… che aspetto I file raw… che aspetto hanno?hanno?Per esempio, da un gas poco strutturato come il Cl2 a 405 K

e con densità numerica (molecolare) di 1.7 molecole/nm-3

(per confronto, un liquido denso può raggiungere densità dell’ordine di 20 molecole/ nm-3) dà profili del tipo:

0 20 40 60 800

2000

4000

6000

8000

ClCl22 sample + sample + cellcell

Empty cellEmpty cell

Cell + Cell + 33He He (matched)(matched)

Empty Empty furnacefurnace

Bragg peak Bragg peak from the from the

cell….cell….

Feeble sign Feeble sign of of

STRUCTURESTRUCTURE

A spike in all A spike in all measurementsmeasurements

(some problem at that detector (some problem at that detector tube…)tube…)

A slightly A slightly different different

slopeslope

Sign of Sign of DIRECT DIRECT beambeam

2 [degrees]

Intensity[arb. units]

I file raw del campione di I file raw del campione di riferimento… riferimento… (H(H22, per esempio. …, per esempio. …

non Vanadio!)non Vanadio!)

0 20 40 60 80 1000

1000

2000

3000

4000

5000

2 [degrees]

Intensity[arb. units]

Dilute HDilute H22

(inside the cell!)

Sign of Sign of DIRECT DIRECT beambeam

0 20 40 60 80 1000

2000

4000

6000

8000

2 [degreces]

Intensity[arb. units]

ClCl22 sample - sample - IIHeHeIIss

(1)(1)

Dopo un po’ di correzioni….Dopo un po’ di correzioni….

Dopo normalizzazione e Dopo normalizzazione e correzionecorrezione

anelasticaanelastica

0 1 2 3 4-2

0

2

4

6

8

Q [Å-1]

D(Q)barn/sr

~