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Neutron Spin Echo Spectroscopy

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Neutron Spin Echo Spectroscopy Neutron Spin Echo Spectroscopy Catherine Pappas TU Delft uncluding slides and animations from R. Gähler, R. Cywinski, W. Bouwman Berlin Neutron School - 30.3.09 from the source to the detector sample and source moderator PSΕ ΙΝ sample environment PSE OUT detector Neutron flux φ = Φ η dE dΩ / 4π source flux intensity field of neutron distribution losses instrumentation definition of the beam : Q, E and polarisation Berlin Neutron School - 30.3.09 Neutron Spin Echo why ? very high resolution how ? using the transverse components of beam polarization Larmor precession Berlin Neutron School - 30.3.09 Neutron Spin Echo I: polarized neutrons - Larmor precession II: NSE : Larmor precession III: NSE : semi-classical description IV: movies V: quantum mechanical approach VI: examples VII: NSE and structure Berlin Neutron School - 30.3.09 Neutron Spin Echo I: polarized neutrons - Larmor precession II: NSE : classical description III: NSE : quantum mechanical description IV: movies V: NSE and coherence VI: exemples VII: NSE and structure Berlin Neutron School - 30.3.09 Polarized Neutrons ๏ polarizer ๏ analyzer magnetic field (guide - precession) magnetic field Neutron Neutron source polarizer B P analyzer detector flipper sample Berlin Neutron School - 30.3.09 Longitudinal polarization analysis Why longitudinal ???? after F. Tasset because we apply a magnetic field and measure the Berlinprojection Neutron School of - 30.3.09 the polarization vector along this field Larmor Precession Motion of the polarization of a neutron beam in a magnetic field dµ = γ (µ B )=µ ω dt · × × L z H “gyromagnetic ratio” µ of neutrons θ y γ =2.9 kHz/G x Berlin Neutron School - 30.3.09 Larmor Precession B S sinθ B S sinθ2 · 1 · dS1 dS dS S = γ (S B ) 2 θ1 dt · × S θ2 dS 1 dS2 ⇒ precession around B ⇒ precession frequency is constant; in both cases dS S sinθ during dt, the angular change of∝ S sin θ around B is constant: Berlin⇒ Neutron the precession School - 30.3.09 ‘Larmor’ frequency ωL does not depend on θ Why Precession relation spin - magnetic moment nucleons electrons e is the elementary charge, is the reduced Planck's constant, mp is the proton rest mass me is the electron rest mass The values of nuclear magneton The values of Bohr magneton SI 5.050 × 10-27 J·T-1 SI 9.274 × 10-24 J•T-1 CGS 5.050 × 10-24 erg•G-1 CGS 9.274 × 10-21 erg•G-1 ratio ~ 1800 Berlin Neutron School - 30.3.09 Larmor Precession NMR spin echo dµ = γµ H = µ ω L Erwin Hahn 1950 dt − × × Berlin Neutron School - 30.3.09 Larmor Precession NMR spin echo dµ = γµ H = µ ω L Erwin Hahn 1950 dt − × × Berlin Neutron School - 30.3.09 Larmor Precession NMR spin echo Neutron spin echo Erwin Hahn 1950 Ferenc Mezei 1972 Berlin Neutron School - 30.3.09 Larmor Precession Neutron spin echo Ferenc Mezei 1972 Berlin Neutron School - 30.3.09 Neutron Spin Echo I: polarized neutrons - Larmor precession II: NSE : Larmor precession III: NSE : semi-classical description IV: movies V: quantum mechanical approach VI: examples VII: NSE and structure Berlin Neutron School - 30.3.09 Larmor precession and flippers Bflipper ⊥ P Polarized neutron beam P P B B d Precession angle φ = ωL t = γ B t = γ B d /v φ = 2916.4 Hz/G ∗ 2π ∗ B [G] ∗ d[cm] ∗ λ [Å] / 395600 For φ = 2 π B d = 135.7 G.cm/ λ [Å] = 1.357 10-3 T.m/ λ [nm] B polariser flipper analyser I+=Io no flip I =I /2 π/2 o Neutron π/2 flip Neutron detector source π flip I-=0 Io -3π -π π 3π Beam Polarization I I+ I 0 P = − − 2 | | I+ + I intensity − -2π 0 2π 0 Δφ ∝Δλ -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 current in the flipper coil [A] neutron spin echo analyser Polarised P ⊥ H sample P // H neutron beam B1 B2 π/2 flipper π flipper π/2 flipper 1 2 o scat. PNSE PNSE 1.357 10-3 T.m/ λ [nm] P//B =〈cos(γ B/v)〉 = ∫ f(v) cos(γ B/v)dv neutron spin echo π flipper P // B P ⊥ B P // B B1 B2 π/2 flipper π/2 flipper 1 2 analyzer polarized neutron beam echo condition does not depend on the wavelength B1 1 = B2 2 Berlin Neutron School - 30.3.09 basic equations S +B -B v φ =0 1 φ1 v2 φ2 φ = ω /v φ = ω (1/v 1/v ) 1 L · 1 2 L · 1 − 2 for v1 = v and v2 = v + ∆v ⇒ φ = ω [1/v 1/(v + ∆v)] ω ∆v/v2 = ω t ∆v/v 2 L − ≈ L L Berlin Neutron School - 30.3.09 basic equations S +B -B v v + ∆v φ =0 φ1 φ2 scattering theory : ∆v ω 2 2 → ω = m/2 (v v )=m/2 (v1 + v2)(v1 v2) m v ∆v · 1 − 2 · − ≈ · · φ =φω =φ[1/v= t 1/(vω+ ∆v)] ω ∆v/v2 = ω t ∆v/v ⇒2 2 L −NSE · ≈ L L 3 and tNSE = ωL /(mv )=ωL t/(2ωo) · · · Berlin Neutron School - 30.3.09 analyser Polarised P ⊥ H sample P // H neutron beam B1 B2 π/2 flipper π flipper π/2 flipper 1 2 o scat. PNSE PNSE scat. S(Q, ω) cos[t (ω ω )] dω P = P cos(φ φ ) = P − o NSE s s S(Q, ω) dω − for quasi-elastic scattering ωo = 0 P scat /P = [S(Q, t)]/S(Q)=I(Q, t) NSE s most generaly φ φ = f(q, ω) S((Q),t) − locally ∝ paramagnetic scattering : P P P echo ⊥ φ P ⊥ 2 −φ P// −P Q Q − P⊥ 2 // P 180 the π flipper is the sample Berlin Neutron School - 30.3.09 measuring principle S(Q, ω): probability for a momentum change Q and an energy change ω upon scattering distribution of spins at analyzer dI/dϕ small tNSE typical S(Q, ω) ϕ = tNSE⋅ ω large tNSE ω ϕ Polar diagrams ϕ P t = t NSE 1 1 P1 t = t NSE 2 P2 0 tNSE t1 t2 Berlin Neutron School - 30.3.09 Fourier transform and resolution Let S(Q, ω) be the scattering function and let R(Q, ω) be the resolution function. Ι If a spectrometer is set to ω’, R(Q, - ’) ω ω then the norm. countrate is: 1 I(Q,ω’) = ∫S(Q, ω) R(Q, ω- ω’) dω S(Q, ) ω = S ⊗ R; convolution; The function R(Q, ω- ω’) should be ω’ ω the same over the range, I where S(Q, ω) is significant; If, like in spin echo, the Fourier Transform of I is the signal, then the convolution of S and R can be written as FT (I) = FT{S ⊗ R} = FT{S}⋅ FT{R} 2 2 γ γo γt γot (γ+γo)t For NSE: FT(I) =FT{ 2 2 ⊗ 2 2 } = e− e− = e− Berlin Neutron School - 30.3.09γ + ω γo + ω Subtleties of NSE ✦ fields are mostly longitudinal; ✦ adiabatic transitions at ends; ✦ ‘π/2 coils’ to start or end precession; ✦ ‘π coils’ to reverse effective field direction; (Hahn’s echoes; NMR-imaging); ✦ ‘Fresnel coils’ to compensate field inhomogeneities ✦ Adiabaticity parameter; at sample; at coils; ✦ How to measure polarization? ✦ Spin flip due to spin-incoherent (2/3) or paramagnetic scattering (depends on O(P, Q )) ✦ Spin echo for ferromagnetic samples; Fresnel coil Fresnel coils: L Standard cylindrical coil: 2 ∫Bdl ≈ B0 L + B0 r /2D; r Correction by current loops: ∫ BF df ∼ I ; density of loops 2 D increases with r ; Berlin Neutron School - 30.3.09 Current around loop Neutron Spin Echo I: polarized neutrons - Larmor precession II: NSE : Larmor precession III: NSE : semi-classical description IV: movies V: quantum mechanical approach VI: examples VII: NSE and structure Berlin Neutron School - 30.3.09 Plane wave entering a static B-field x σ+ σ+ ⇒ H? B = Bz z y σ- σ- y = 0 the 2 states Ψ+ and Ψ- have different kinetic energies E0 ± µ⋅B Static case [dB/dt = 0]: no change in total energy (ω = ω0) but change in k µ⋅B ¸ Ekin ⇒ Berlin Neutron School - 30.3.09v is the classical neutron velocity Ψ ei(yk+ ωot) e+iy∆k Ψ = + = − i(yko ωot) i(yk ωot) = e − iy∆k Ψ e −− e− − Both states have equal amplitudes, as the initial polarization is perpendicular to the axis of quantization (z-axis); 1 These amplitudes are set to 1 here. √2 Energy diagram: Ekin [-E ] pot H= ⎡ 1 ⎤ H H = H ⋅ ei Δk y 1 0 + 0 ⎣ ⎦ B E0 µ Δk = µ ⋅B/v 2 –i Δk y H- = H0 ⋅ e Berlin Neutron School - 30.3.09y=0 y in a magnetic field Ekin [-Epot] E0 B B = B µ Z 2 0 y0 y Setting the polarizer to x-direction: Δk = µ ⋅B/v t0 = y0 /v Berlin Neutron School - 30.3.09 Larmor precession! wavepackets instead of plane waves wavepacket (bandwidth Δλ) of length Δy and lateral width Δx = Δz ; Δy ≈ λ2/ Δλ; Δx ≈ λ/(2πθ); θ = beam divergence; typ. values: Δx, Δy ≈ 100 Å Δk = µ ⋅B/v; v δ = v·t y0 Time splitting dt = t of the two wave packets, separated by the propagation through the field of length y0: dE 2t y 2µB 2µB y ω t = t = o = o = L 1 2 ·3 · E to ⇒ 2 · m m 2ωo ∨ 2 ∨ ∨ For fields of typ. 1 kG and length of m, t is in the ns range for cold neutrons; Berlin NeutronIn Neutron School - 30.3.09Spin echo spectroscopy, t is the ‘spin echo time’; semi-classical description: measuring principle Ekin E0 + µB E0 + µB E0 B S -B analyzer E0 - µB E0 - µB ✤The first field splits the wavepacket into two ✤the second one overlaps them again; ✤The analyser superposes both packets; Complete overlap of v tNSE scattered wave packets S Berlin Neutron School - 30.3.09 Neutron Spin Echo I: polarized neutrons - Larmor precession II: NSE : Larmor precession III: NSE : semi-classical description IV: movies V: quantum mechanical approach VI: examples VII: NSE and structure Berlin Neutron School - 30.3.09 S 104 103 102 101 100 10-1 S(Q,τ) 104 length d = 2π/Q [Å] VUV- Inelastic 10-2 102 FEL x-ray Raman Chopper scattering UT3 0 100 10 ] Multi-Chopper time t [ps] timet red Infra- Brillouin Inelastic Neutron scattering 2 -2 10 [meV 10 Scattering ν Backscattering 104 10-4 Spin Echo µSR NMR Photon 6 correlation 10 energyh E = 10-6 X-ray correlation spectroscopy spectroscopy Dielectric 108 10-8 10-4 10-3 10-2 10-1 100 101 102 -1 scattering vector Q [Å ] source: ESS Berlin Neutron School - 30.3.09 Neutron Spin Echo I: polarized neutrons - Larmor precession II: NSE : Larmor precession III: NSE : semi-classical description IV: movies V: quantum mechanical approach VI: examples VII: NSE and structure Berlin Neutron School - 30.3.09 what is the real life - what is fantasy ? semi-classical picture: v after ‘short field’ after ‘long field’ Δy Δy ≈ λ2/ Δλ; Larmor precession Directions of the individual spins of the polychromatic beam after passage through B fields of different lengths Complete separation of the two packets implies that no coherent superposition of both states exists any more Berlin Neutron School - 30.3.09 what is the real life - what is fantasy ? what is a spin ? what is the coherent superposition of states ? quantum mechanics = plane waves Berlin Neutron School - 30.3.09 is a coherent superposition of the two states their “sum” ? + + vs.
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