Echo

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 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 - 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 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￿ = γ￿µ H￿ = µ￿ ￿ω L Erwin Hahn 1950 dt − × ×

Berlin Neutron School - 30.3.09 Larmor Precession

NMR spin echo dµ￿ = γ￿µ 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 ω￿o = 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 and resolution

Let S(Q, ω) be the scattering function and let R(Q, ω) be the resolution function.

Ι If a 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 ⎡1⎤ i Δk y H= H H+ = H0 ⋅ e ⎣1⎦ 0 E0 Δk = µ ⋅B/v B 2 µ –i Δk y H- = H0 ⋅ e

Berlin Neutron School - 30.3.09y=0 y in a magnetic field

Ekin [-Epot]

E0 B = BZ B 2 µ

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 energy E = h energy 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. the spin in the plane | ￿ |−￿ the spin is not a vector (classical view) what is classical what is quantum mechanical

Berlin Neutron School - 30.3.09 coherent superposition of states the spin is not a vector (classical)

s=1/2 +1/2 S sz= -1/2

Berlin Neutron School - 30.3.09 coherent superposition of states

spin 1 case gives a hint: two sz=0 states

Berlin Neutronsinglet School - 30.3.09 sz=0 of the S=1 state coherent superposition of states

Basics of QM : 2 slit experiment wave - particle duality The classical - “Copenhangen” - description of QM

The Problem of Measurement E. Wigner 1963

Berlin Neutron School - 30.3.09 coherent superposition of states

Basics of QM : 2 slit experiment wave - particle duality The classical - “Copenhangen” - description of QM

The Problem of Measurement E. Wigner 1963

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 -

right: π /2 flipper coils above: Mezei’s first spin echo precession coils HZ Berlin

below: IN11 Examples of neutron spin echo Reptation in polyethelene studies The dynamics of dense polymeric systems are dominated by entanglement efects which reduce the degrees of freedom of each chain

de Gennes formulated the reptation hypothesis in which a chain is confined within a “tube” constraining lateral difusion - although several other models have also been proposed

The measurements on IN15 are in agreement with the reptation model. Fits to the model can be made with one free parameter, the tube diameter, which is estimated to be 45Å

Schleger et al, Phys Rev Lett 81, 124 (1998) 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 codes scattering angle

θ sample

Unscattered beam gives spin echo at φ = 0 Independent of height and angle

Scattered wave vector transfer Qz

results in precession φ φ ≅ zQz Proportional to the spin echo length z

Measure polarisation: P(z) = cos(zQz) Fourier transform scattering cross-section: real space density correlation function G(z)

Keller et al. Neutron News 6, (1995) 16 Rekveldt, NIMB 114, 366 (1996). Direct information in measurement

Ordering Compactness Correlation lengths Fractal dimensions Specific surfaces Etc… SESANS measures directly shape ridges Realisation SESANS

monochromatorpolariser magnet 1 field stepper guide field analyser detector sample polariser polariser Spin-echo reflectometer OffSPEC

• Off-specular to measure in-plane scattering • Specular reflectivity of bent surfaces high-resolution • Separation specular and off-specular

ISIS 2nd target station Off specular reflectometer Spin-echo components for High resolution without collimation Thank you !