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Introduction to Triple Axis Neutron Spectroscopy

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Introduction to Triple Axis Neutron Spectroscopy Introduction to Triple Axis Neutron Spectroscopy Bruce D Gaulin McMaster University The triple axis spectrometer Constant-Q and constant E Practical concerns Resolution and Spurions Neutron interactions with matter Properties of the neutron -27 Mass mn =1.675 x 10 kg Charge 0 Spin-1/2, magnetic moment µn = -1.913 µN • Neutrons interact with… • Nucleus • Crystal structure/excitations (eg. Phonons) • Unpaired electrons via dipole scattering • Magnetic structure and excitations Nuclear scattering NXS School Magnetic dipole scattering 2 Brockhouse and Shull Share 1994 Nobel Prize in Physics Most recent Nobel Prize awarded to a scientist working in Canada The Basic Experiment: (θθθ, φφφ) Incident Beam: Scattered Beam: • monochromatic • Resolve its energy • “white” • Don’t resolve its energy • “pink” • Filter its energy Brockhouse’s Triple Axis Spectrometer | ki | = 2 π / λi | kf | = 2 π / λf Momentum Transfer : Q = ki – kf Q ki - kf Energy Transfer: k f 2 2 2 δ E = h /2m (k i – kf ) Mapping Momentum – Energy (Q-E) space Origin of reciprocal space; Remains fixed for any sample rotation 2 πππ / a 2 πππ / a Bragg diffraction: Elastic scattering : | ki | = | kf | Constructive Interference Q = Reciprocal Lattice Vector kf k i -kf Q Bragg diffraction: a Constructive Interference Q = Reciprocal Lattice Vector kf ki -kf Q 2 πππ / a Elastic scattering : | ki | = | kf | Elementary Excitations in Solids • Lattice Vibrations (Phonons) • Spin Fluctuations (Magnons) Energy vs Momentum • Forces which bind atoms together in solids Bragg’s Law: nλλλ = 2d sin( θθθ) Bragg’s Law: nλλλ = 2d sin( θθθ) Two Axis Spectrometer : • 3-axis with analyser removed • Powder diffractometer • Small angle diffractometer • Reflectometers Diffractometers often employ working assumption that all scattering is elastic. Soller Slits: Collimators Define beam direction to +/- 0.5, 0.75 etc. degrees Filters: Remove λλλ /n from incident or scattered beam, or both Single crystal monochromators: Bragg reflection and harmonic contamination nλλλ = 2d sin( θθθ) Get: λλλ , λλλ/2 , λλλ/3 , etc. Pyrolitic graphite filter: E = 14.7 meV λ = 2.37 A v = 1.6 km/s 2 x v = 3.2 km/s 3 x v = 4.8 km/s Constant k f Constant k i Two different ways of performing constant-Q scans Mapping Momentum – Energy (Q-E) space Origin of reciprocal space; Remains fixed for any sample rotation 2 πππ / a 2 πππ / a Spurions • Bragg – incoherent – Bragg 6000 500 (350,4) (mHz,S/N) – Eg. ki – 2k f • ħω = 41.1 meV 400 • E = 13.7 meV 300 (150,2) f (90,1) (40,0.3) Ei = 54.8 meV 200 4E f = 54.8 meV 100 NeutronNeutronCounts/1000s Counts/1000s • Incoherent elastic scattering 0 visible from analyzer λ/2 0 20 40 60 80 100 Energy Transfer (meV) • incoherent – Bragg – Bragg – Sample 2 θ in Bragg condition for kf-kf – Even for inelastic config, weak incoherent from mono May 31, 2009 NXS School 22 Elementary Excitations in Solids • Lattice Vibrations (Phonons) • Spin Fluctuations (Magnons) Energy vs Momentum • Forces which bind atoms together in solids Constant Q, Constant E 3-axis technique allow us to Put Q -Energy space on a grid, And scan through as we wish Map out elementary excitations In Q-energy space (dispersion Surface) Phonons Normal modes in periodic crystal wavevector 1 u(l,t) = ∑εεε ()q exp ()iq⋅ l Bˆ ( qj,t) NM j jq Energy of phonon depends on q and polarization FCC structure LongitudinalTransverse mode mode NXS School FCC Brillouin zone 25 Lynn, et a l., Phys. Rev. B 8, 3493 (1973). Phonon intensities Structure (polarization) factor 2 1 −Q 2 u2 Q ⋅εεε j ()q S1+ ()Q,ω = e ∑ ()1+ n()ω δ()Q − q −τττ δ( ω −ω j ()q ) 2NM jq ω j ()q Long Trans PhononPhononEnergy Energy 0q, [ ξ00] 1/2 NXS School 26 Guthoff et al., Phys. Rev. B 47 , 2563 (1993). More complicated structures Acoustic phonon Woods, et al., Phys. Rev . 131 , 1025 (1963). Optical phonon La 2CuO 4 NXS School 27 Chaplot, et al., Phys. Rev . B 52 , 7230(1995). Detectors • Gas Detectors • n + 3He 3H + p + 0.764 MeV • Ionization of gas • e- drift to high voltage anode • High efficiency • Beam monitors • Low efficiency detectors for measuring beam flux NXS School 28 Resolution • Resolution ellipsoid – Beam divergences – Collimations/distances – Crystal mosaics/sizes/angles • Resolution convolutions I( Q0,ω0 ) =∫ S( Q0,ω0 )R(Q − Q0,ω −ω0 )dQdω NXS School 29 Resolution focusing • Optimizing peak intensity • Match slope of resolution to dispersion May 31, 2009 NXS School 30 References General neutron scattering G. Squires, “Intro to theory of thermal neutron scattering”, Dover, 1978. S. Lovesey, “Theory of neutron scattering from condensed matter”, Oxford, 1984. R. Pynn, http://www.mrl.ucsb.edu/~pynn/. Polarized neutron scattering Moon, Koehler, Riste , Phys. Rev 181 , 920 (1969). Triple-axis techniques Shirane, Shapiro, Tranquada, “Neutron scattering with a triple-axis spectrometer”, Cambridge, 2002. Time-of-flight techniques B. Fultz, http://www.cacr.caltech.edu/projects/danse/ARCS_Book_16x.pdf NXS School 31 Constant kf: • kf , θθθA do not change; therefore analyser “efficiency” is constant • ki, θθθM do change, but monitor detector normalizes to incident neutron flux • Monitor detector (low) efficiency goes like ~ 1/v ~ 1/k i Monochromators • Selects the incident wavevector 2π Q(hkl ) = = 2k sin θ θ d(hkl ) i Q(hkl) • Reflectivity Mono d(hkl) uses • focusing PG(002) 3.353 General • high-order contamination Be(002) 1.790 High k i eg. λ/2 PG(004) Si(111) 3.135 No λ/2 l Guides • Transport beam over long distances • Background reduction • Total external reflection – Ni coated glass – Ni/Ti multilayers (supermirror) NXS School 34 Samples • Samples need to be BIG – ~ gram or cc – Counting times are long (mins/pt) • Sample rotation • Sample tilt HB3-HFIR IN14-ILL Co-aligned CaFe 2As 2 crystals May 31, 2009 NXS School 35.
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