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Diffrazione ad alti angoli su un 2 assi: i problemi, le misure & lanalisi dati Eleonora GUARINI Dipartimento di Fisica, Università di Firenze

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Presentazione sul tema: "Diffrazione ad alti angoli su un 2 assi: i problemi, le misure & lanalisi dati Eleonora GUARINI Dipartimento di Fisica, Università di Firenze"— Transcript della presentazione:

1 Diffrazione ad alti angoli su un 2 assi: i problemi, le misure & lanalisi dati Eleonora GUARINI Dipartimento di Fisica, Università di Firenze Giornate Didattiche 2010 Hotel Steinpent, S. Giovanni in Valle Aurina 27 – 30 Giugno 2010 Società Italiana di Spettroscopia Neutronica

2 Outline What we measure Versu s 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

3 The sought-for quantity The central quantity in the study of the microscopic structure of an isotropically scattering sample (a liquid, a powder…) is STATIC STRUCTURE FACTOR The accessible quantity On a two-axis diffractormeter (fixed incident energy, angular dispersive) we measure:

4 Origin of the accessible quantity I The double-differential cross section for nuclear neutron scattering (bs are the scattering lengths) is the probe fingerprint While, from the theory of space- and time-dependent correlation functions, it is found: Lets make the resemblance between double-differential cross section and S(Q,) more evident as: Were the scattering lengths (i.e. the fingerprint of the probe we are using) absent, this would be exactly the dynamic structure factor with

5 Coherent and incoherent components in neutron scattering The scattering lengths b depend on the specific isotope, and on the relative orientation of nuclear and neutron spin Coherent scattering is related to the average value of b Incoherent scattering is related to the spread (variance) of the bs distribution

6 beamdetectorsample Origins II… (in the ideal case ) 2 k0k0 k1k1 2 Q k0k0 k1k1 Restrictions: no magnetic effects, no polarizations. Assumption: fixed incident energy E 0 Energy analysis of emerging neutrons (, ) In the ideal case ALL relevant quantities are exactly defined A 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

7 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: This quantity differs EVEN from the differential cross-section, which is defined as: related to the probe fingerprint because of the detectors efficiency dependence on the energy of the scattered neutrons.

8 BUT, if inelasticity effects are SMALL…. i.e., … if E 0 >> E, then: and also: per un monoatomico e monoisotopico

9 Dreaming a little more... idealized diffraction experiment The idealized diffraction experiment would require the combination of: - a perfect instrument - an ideal sample - a direct I( 2) S(Q) relation the perfect diffractometer the ideal sample No background Uniform and collimated beam Perfect k 0 definition High angular resolution Detector efficiency 100% Some 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…

10 Towards the real case.I. Beam attenuation absorption scattering 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. absorption and scattering. slab sample Consider the simplest case where a uniform, collimated, and monochromatic beam crosses a homogeneous slab sample perpendicular to the beam axis: 0 L dx x I0I0 What is T ? 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) Thus,

11 Towards the real case.II. Transmission & Attenuation coefficients 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 0 x 2 P so that 0° 120° 2 A(2 ) Liquid in a cylindrical container

12 Towards the real case.III. Multiple scattering If a neutron is scattered once, it can be scattered 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: single scattering intensity It removes singly scattered neutrons from the original direction. Therefore it ATTENUATES the single scattering intensity detectable at a given angle another true single scattering It contributes to the intensity detected at another angle. Therefore it INTENSIFIES the signal, mixing up with the true single scattering component at a given angle. Multiple scattering causes both the loss of good neutrons and the detection of bad ones! and the detection of bad ones! Generally, one can write: The multiple scattering intensity can be evaluated, starting from the double contribution, by Monte Carlo integration. Different degrees of approximation are possible.

13 2 k 1 + k 1 k 0 + k 0 2 e The real-life situation Real instrument Real sample,MS

14 Instrumental effects.I. Background noise It 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 I EB (,) which approximates the thorough instrumental noise. 2) A run with a full absorber (typically Cadmium) in place of the sample+container system. The collected I Cd (,) 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

15 Instrumental effects.II. Detector efficiency It 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) geometry c) 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 n D and thickness L: For a cylindrical detector perpendicular to the beam, with radius r and density n D : r y 0 dy x L(y)L(y) r

16 Sample-related effects.I. Attenuation Due to absorption and multiple scattering in the sample we have A s,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 E 1 ) BUT INELASTIC EFFECTS ARE TYPICALLY NEGLECTED IN THE EVALUATION OF THE ATTENUATION The expression of A s,s (2 ) can however be derived by taking geometrical and size effects into account...

17 Attenuation & size-effects Lets write our best expression for I s (1) exp (2 ): with = AND L = L(P,D) = (P,D) true = true (P,D) Paalman&Pings If we neglect the (weak) dependence of 2 true 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&Pings " coefficient including size-effects : S d /L D 2 θ true L D θ LDLD P dS d 0 1

18 Correction for Attenuation 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 needs Level 2 : maximize the incident energy, compatibly with the property under study (your wavelength should fit your d-spacings!!) A posteriori correction A posteriori correction: Expt. or Calc. For each 2 of the experiment, calculate A s,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 you need a good model for the unknown double differential cross section…) To simplify things, it is often assumed that: A s,s (2 ) A s,s ( = 0, = 0 )= T s

19 Complication: the container In 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,cc,scc,c s,sc

20 Correction for attenuation in the presence of a container A priori experimental correction A priori experimental correction (minimization): A posteriori correction A 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 A c,c (2 ), A c,sc (2 ) and A s,sc (2 ) !!! Example: Monte Carlo integration (P,D) couples are randomly sampled over the 5-dimensional space (V c ill + S d ) (in the elastic approximation)

21 PAUSA?????

22 Sample-related effects.II. Multiple scattering Due to multiple scattering in the sample we have But we cant measure I s (m)exp and we dont know the scattering law. L θ θ1θ1 θ2θ2 LDLD D P1P1 dS d P2P

23 Correction for Multiple Scattering A priori experimental correction A priori experimental correction (minimization): A posteriori correction A 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) simulate Given a model, a way is to simulate multiple scattering processes calculate Alternatively, calculate by Monte Carlo integration (P 1,P 2,D) triplets are randomly sampled over the 8- dimensional space (V s ill + V s + S d ). (in the elastic approximation) (P 1,P 2,D) triplets are randomly sampled over the 8- dimensional space (V s ill + V s + S d ). (in the elastic approximation)

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

25 Complication: multiple scattering involving the container cc,ccc,sccc,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,sccs,sc

26 Multiple scattering correction in a multi-element system 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 ) But we skip the details…..

27 Handling experimental quantities and calculations We have: EXP with

28 The single-scattering intensity I s (1) If multiple scattering is neglected If attenuation coefficients are approximated by the transmissions (A s,sc A c,sc T sc T s T c and A c,cT c ) F is the instrumental factor due to flux, solid angle, detector efficiency and sample density … we need to MEASURE it

29 Data normalization to absolute 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… and their 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

30 Data normalization to absolute units 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:

31 The data-analyzer endless enigma How much should one push the refinement of the data analysis and related calculations? an extremely accurate analysis would still be approximate and would never end! It depends on… The physical effect under investigation The accuracy of the neutron experimental data and available TIME Unavoidable measurements Background Empty container Reference sample (Vanadium) Unavoidable estimates attenuation multiple scattering Various degrees of accuracy the sample features may allow for reasonable approximations down to: use of transmissions neglect of multiple scattering Useful measurements Transmissions

32 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. Lintensità 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 durto differenziale NON è ciò che si cerca perché contiene ancora le sezioni durto 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)

33 Facciamo finta di arrivare sul D1A di ILL… … e di dover fare delle misure su un liquido. A priori avrete scelto un contenitore adatto al vostro liquido in termini di: Potere di scattering del campione corrispondente… P s 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…)

34 Facciamo finta di arrivare 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 cella di Cadmio (prendete vari riferimenti geometrici per poterla riposizionare bene, ma solo con Cd esterno)

35 Facciamo finta di … A posteriori (in genere, ma dipende…): Va scelta lenergia 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 strumento accuratezza che si desidera raggiungere sullintensità 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….

36 I file raw… che aspetto hanno? Per esempio, da un gas poco strutturato come il Cl 2 a 405 K e con densità numerica (molecolare) di 1.7 molecole/nm -3 (per confronto, un liquido denso può raggiungere densità dellordine di 20 molecole/ nm -3 ) dà profili del tipo: Cl 2 sample + cell Empty cell Cell + 3 He (matched) Empty furnace Bragg peak from the cell…. Feeble sign of STRUCTURE A spike in all measurements (some problem at that detector tube…) A slightly different slope Sign of DIRECT beam 2 [degrees] Intensity [arb. units]

37 I file raw del campione di riferimento… (H 2, per esempio. …non Vanadio!) 2 [degrees] Intensity [arb. units] Dilute H 2 (inside the cell!) Sign of DIRECT beam

38 2 [degreces] Intensity [arb. units] Cl 2 sample - I He I s (1) Dopo un po di correzioni….

39 Dopo normalizzazione e correzione anelastica Q [Å -1 ] D(Q) barn/sr ~

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