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Basics on Diffraction

Basics on Diffraction. Pierre Becker Ecole Centrale Paris SPMS CNRS Lab pierre.becker@ecp.fr. New position : Augure of Jupiter Calchas. « Le cornichon ( piclke ) est confit dans du vinaigre, je suis confident du Roi ».

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Basics on Diffraction

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  1. Basics on Diffraction Pierre Becker Ecole Centrale Paris SPMS CNRS Lab pierre.becker@ecp.fr New position : Augure of Jupiter Calchas « Le cornichon (piclke) est confit dans du vinaigre, je suis confident du Roi » JDN 22, September 21-24 2014

  2. Scattering and diffraction of angstromwavelengthwaves (X Ray, electrons, neutrons) is the key to study of atomic structure of condensedmatter. High resolutionscatteringallows for understandingvibrational, electronic and magneticbehaviour of materials: key importance of combined X, N, E studies New pulsed sources allow for a time dependentstudy for out of equilibriumsystems I shall focus on neutron scattering, but keep in mind the importance of combinedstudies JDN 2014 21-24 septembre Oleron

  3. Outline • Nuclear Diffraction by a stationarytarget • - Case of a crystal. ComparisonwithXray and Electron diffraction • - Taking thermal motion intoaccount : total/Bragg scattering • Inelastic neutron scattering and study of phonons • Magneticscattering of neutrons • Basics • Spin polarized diffraction: access to spin density of materials: complementaritywithXrayhighresolutionscattering • Reality of crystals: extinction JDN 2014 21-24 septembre Oleron

  4. F0: incident flux – F: scattered flux Scattering cross section X Rays, electrons, neutrons JDN 22, September 21-24 2014

  5. Case of energy sensitive detection • Non relativistic particles • Photons: prefactor includes polarization factor of EM beam JDN 22, September 21-24 2014

  6. Elementaryscatterer: Scatteringby a stationarytarget: Huyghens Fresnel principle: elastic scattering JDN 22, September 21-24 2014

  7. P r k f r q Q=4psinq/l q r k 0 Long distance scattering JDN 22, September 21-24 2014

  8. Complextarget: Scatteringfunction: JDN 22, September 21-24 2014

  9. Crystallinetarget (periodicscatteringfunction) One defines the structure factor : For a finitecrystal, with N unit cells Coherentscattering . JDN 22, September 21-24 2014

  10. Bragg diffraction conditions. is the momentumtransferred to the target by the wave. For a finitecrystal, thereis an opening angle around Bragg condition, of order d/L and one must integrate the peakintensity Notice that the scatteredwave has a phase shift of p/2with respect to the incident wave JDN 22, September 21-24 2014

  11. Case of X Rays Elementaryscatterers are electrons, all undiscernable For a 1A0 wavelength, the energy of the photon is about 12000ev, muchhigherthancohesiveenergies in a material Electron charge densityis the effective « target » : Independent atom approximation JDN 22, September 21-24 2014

  12. Formfactorsatlowscattering angle are proportional to atomicnumber Theydecay as the inverse of atomic radius JDN 22, September 21-24 2014

  13. Neutron waves Neutrons result from fission of U235 and are produced with an energy > 1Mev Moderation via collisions with D2 O, liquid hydrogen, graphite Neutron life time is about 900 sec Also spallation sources, producing pulsed neutrons Wave-particleduality For l =1 Å E =80 meV T = 950K v = 4000 ms-1 Neutron is a fermion, with ½ spin JDN 22, September 21-24 2014

  14. One talks about « thermal Neutrons », behavinglike a perfect gaz flow It isnecessary to monochromatise the beam, l ≈ l0 , whichalsoproducessignificantbeamwith l/2 wavelength The speed of neutrons islowenough to allow for time of flight detection, besidesusualselective absorption JDN 22, September 21-24 2014

  15. Basic interaction withmatterisnuclear interaction, which for study of materials, canbeconsidered as ponctual (range of fm) b iscalled the scatteringlength by a nucleus, of the order of 10-12 cm VerysignificantcomplementaritytowardsXRays JDN 22, September 21-24 2014

  16. Let’sconsider an incident neutron wavedirectedtowards a nucleus locatedatorigin. Whenreaching the static nucleus, the neutron wavesatisfies the Schrodingerequation It canbere-written as After collision, the neutron isscatteredwith the sameenergy, and canbedescribed as a sphericalwave. One can show (1rst Born approximation, sinceb isverysmallcompared to l) that the proper solution (Green function) is JDN 22, September 21-24 2014

  17. Complextarget JDN 22, September 21-24 2014

  18. Generalization to electronscattering and beyond It willalsoapply to magnetic interactions of neutrons withmatter Important complementarity of Xray / Electron scattering JDN 22, September 21-24 2014

  19. For a stationarycrystal, forgetting about surface effects… Phase problemremains to retrieve the scatteringdensity Only the Patterson functioncanbeobtaineddirectlyfromexperiment JDN 2014 21-24 septembre Oleron

  20. Also direct methods are available for getting the phases Great thanks to David Sayre, JeromeKarle and Herbert Hauptman JDN 2014 21-24 septembre Oleron

  21. Due to finite size, scatteringoccurs out of exact Bragg condition For a given position of the crystal (e), the cross section per unit volume iss(e), and after rotation of the crystal, one gets the total « kinematic » diffraction cross section per unit volume : e Raies de surstrucutre (1/2 1/2 1/2) pour PbMg1/3Nb2/3O3 (PMN), à T = 300 K et 10 K, le long de la direction [hhh] spectromètre 6T2 LLB JDN 2014 21-24 septembre Oleron

  22. Diagramme de diffraction des neutrons du BaTiO3 à 401 K : spectromètre 3T2 LLB JDN 2014 21-24 septembre Oleron

  23. There are resolutionlimits due to the factthat There canbe a mixture of isotopes in a compound, and moreovertherecanbe a spin couplingbetween the neutron and scattering nucleus. As a consequence, the scatteringlength of a givenatomcanvaryfrom one site to another one The total cross-section iswritten as the sum of a coherent and an incoherent part JDN 2014 21-24 septembre Oleron

  24. Isotopicincoherence for Ni Spin incoherence for H sc si H 1,8 81,2 D 5,6 2 O 4,2 0 V 0,02 5,0 Ni 13,4 5,0 One gets, besides a coherent Bragg diffraction, where the appropriatescatteringlength for a nucleus is<bj> an incoherentaddedintensity Absorption canalso damage the signal, but muchlessthan for X Rays, Only few atoms have a significant absorption cross section. It cancels for polarized neutron data JDN 2014 21-24 septembre Oleron

  25. Diffraction experimentsoccur on systemsat a giventamperature. This createsvibrationaldisorder of atomic positions. In crystals, those vibrations are phonons Bragg diffraction Thermal diffuse scattering Debye Waller factor In the harmonic approximation, one can show exactly JDN 2014 21-24 septembre Oleron

  26. For a givenfrequency Debye Waller factor lowers signal for high T, large scattering angle, light atoms, with major contribution fromacousticfrequencies. Anharmonic motion canalsobeconsidered For systems made of molecules or for instance organic fragments linked to a protein, one can model vibrations in terms of intra-molecular (highfrequency) and inter-molecularacoustic vibrations ! Staticatomicdisordercanalsolead to a lowering factor, not dependent on T (see extinction !!!) The otherterm <DA2> corresponds to thermal diffuse scattering. Seelaterit’sfrequencyanalysis. If no energyanalysisisdone, itis a diffuse contribution, whichisconsidered as part of background JDN 2014 21-24 septembre Oleron

  27. Absorption of thermal neutrons JDN 2014 21-24 septembre Oleron

  28. Parametersprogressivelyintroduced in the structure factor and the scatteringdensityfunctioncanbeadjusted by a refinement of observed data towards the proposed model, via a minimisation with respect to adjustablekeyparameters (positions, thermal parameters…) Anotherapproachis the maximum entropymethod Besidescrystals, many applications to aperiodic structures, disorderedsolids (glass) and liquids Model refinementcanbeverycritical JDN 2014 21-24 septembre Oleron

  29. Quantum approach Fermi Golden Rule JDN 22, September 21-24 2014

  30. Space and time correlationfunction JDN 2014 21-24 septembre Oleron

  31. Most general situation System and beam in a statistical initial state JDN 22, September 21-24 2014

  32. JDN 22, September 21-24 2014

  33. Static approximation Total scattering Mostlyvalid for X Rays JDN 22, September 21-24 2014

  34. Elasticscattering Total scattering JDN 22, September 21-24 2014

  35. Phonon scattering Considerinelasticscattering for a model crystalwith one atom/cell, where the system will go fromvibrational state yn to ym Displacementcanbedecomposed in phonons. Wesimplify by consideringonly one phonon, of wavevectorq JDN 22, September 21-24 2014

  36. Creation or absorption of phonon Energy change canbeverypronouncedwith neutrons, verysmallwith X Rays (need for veryhighresolution JDN 22, September 21-24 2014

  37. One gets diffraction condition (coherent effect) In a givenscattering direction thus, withenergy analyser, one getsw, kf, and thereforeq, w(q) JDN 22, September 21-24 2014

  38. Key application of neutrons, in particular for phase transitions JDN 2014 21-24 septembre Oleron

  39. Neutron Magneticscattering Magneticscatteringoccursthroughdipolar interaction between neutron and electronmagnetic moments JDN 22, September 21-24 2014

  40. Spin component Spin density JDN 22, September 21-24 2014

  41. Orbital component, related to currentdensity Orbital magnetisationdensitydefined to any gradient, but magnetic amplitude uniquelydefined JDN 22, September 21-24 2014

  42. Normalized spin density Case of spin only magnetism, at zero temperature Magnetizationdensity in units of 2mB., with a quantization axis 0z (appliedfield or naturalquantization axis) Ground state eigenstate of S2 andSz. JDN 22, September 21-24 2014

  43. Often, orbital moment is not a constant of motion. Often, spin orbitcouplingremainslocalized, and Lremains a reasonably good quantum number. (L2, S2, J2, Jz) are the good quantum numbers Orbital magnetization In units of 2mB JDN 22, September 21-24 2014

  44. If no spin analysis for scattered beam, two cross sections Neutron elastic scattering JDN 22, September 21-24 2014

  45. Unpolarized neutrons Magnetic effects hard to observe with unpolarized neutrons JDN 22, September 21-24 2014

  46. Polarized neutron diffraction Incident beam of polarized neutrons (spin ) vertical magnetic field verticalpolarisation down polarising monochromator flipper single crystal JDN 22, September 21-24 2014

  47. simple case : in horizontal plane Magnetization along Oz Flipping ratio with Linear en FM polarised si FM<< FN non polarised negligeable advantage of polarized neutrons for for weak magnetism JDN 22, September 21-24 2014

  48. General case : out of horizontal plane aligned along vertical axis z z component of a y x JDN 22, September 21-24 2014

  49. Strongcomplementaritywith X Ray highresolution diffraction, leading to charge density: mainlycohesive contribution towardssum of independentatoms P. Coppens and I, with T. Koristzansky, initiatedsuchcombinedstudy Presently a strongprojectwith LLD, CRM2, Spring8 and ourlab SPMS combinedrefinement of charge and spin, and also charge / spin and momentum. JDN 2014 21-24 septembre Oleron

  50. Real crystals: extinction JDN 22, September 21-24 2014

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