Neutron Optics and Polarization T

Neutron Optics and Polarization T. Chupp University of Michigan With assistance from notes of R. Gähler; ILL Grenoble

1. Neutron waves 2. Neutron guides 3. Supermirrors break 4. Neutron polarization 5. Neutron polarimetry 6. Neutron spin transport/flipping

General references: V.F. Sears, Neutron Optics, Oxford 1989 Rauch and Werner, Neutron Interferometry, Oxford 2000 Fermi: Nuclear Physics (notes by Orear et al. U. Chicago Press - 1949) QM Text (e.g. Griffiths) Optics Optics: the behavior of light (waves) interacting with matter

Waves characterized by wavelength λ

Matter characterized by permeability κ, susceptibility µ, dissipation (ρ/σ)

Interaction characterized by n (index of refraction); δ (skin depth)

Useful when " >> a (atomic spacing)

! deBroglie: massive particles behave as waves 2" p k = = # h 2 2 2 2 p c 2 4# (hc) k T = " = 2 B 2 2mc k T ! 2mc B mc2 = = 939.6 MeV

! h c = 197.3! MeV-fm = 1973 eV-Å

v = 2200 m/s λ = 1.8 Å (thermal neutrons - 300° K) ! 1 Note also: " # T

! Wave Properties

E • Polarization y x different meaning mirror Vertically polarized light • Reflection i r (angle of incidence = angle of reﬂection)

• Refraction i ni sin i = nt sin t t 8 n=c0/cc0=2.97x10 m/s • Interference Superposition + = m λ=W sin θ • Diffraction The wave equations for light and matter waves in vacuum: 2" p k = = # h EM wave equation (E, B) Schrödinger equation

2 2 1 % # 2 m $# Time dependent: ! " # $ 2 2 = 0 " # + 2i = 0 c %t h $t r r "(r , t) = a ei(k #r $% k t ) k Time independent 2 (rv ) k 2 (rv ) 0 Helmholtz! equation: " #! + # = Schroedinger equation E 2 2mE k 2 = k 2 = Dispersion relations:! ( c)2 2 h h ! " " m2c 2 " c v = = c v = 1+ # Phase velocity: ph k ph k p2 k v ! 2 2 2 4 ! E = p c + m c ( E = h") ( )

! ! ! ! Interactions V(r) 2m " 2#(rv ) + E $V (rv ) #(r) = 0 Time independent 2 [ ] Schroedinger equation h f (#) v v (rv ) eikz eik $r Incoming plane wave " = + Outgoing spherical wave r

! 1 2i # l Partial waves f (") = %(2l +1)[e $1]Pl (cos") 2ik l ! 1 1 f (") = [e2i# 0 $1] = s-wave scattering " = #kr V(r)=0 at r 2ik [k cot#0 $ ik] 0 o o

a=- δ_0_: scattering length ! a 2 2 k f (") = #a + ika + O(k ) (-5 fm < a < 15 fm) ! -4 ka=-δ0 ~ 10 ! f (") # $a !

! Coherent Scattering Lengths b = _A+1____ a A

element b (fm) H -3.74 Be 7.79 C 6.65 Al 3.45 Si 4.15 Ti -3.44 Fe 9.45 Co 2.49 Ni/58Ni 10.3/14.4 Cu/65Cu 7.72/10.6 Cd 4.87-0.7i

1 2i# 0 f (") = [e $1] "0 = #ka 2ik

! ! Index of refraction

2 v 2m v Time independent " #(r ) + 2 [E $V (r )]#(r) = 0 Schroedinger equation h 2 v 2 v " #(r ) + K #(r ) = 0 2m K 2 = E "V(rv ) ! 2 [ ] h c K ! For light: c " = n k

! V (rv ) In general, n is a tensor, i.e. V(r) n(rv ) = 1" depends on propogation direction E ! 2" 2 V(rv ) = h b$ 3 (r % r ) Fermi Pseudopotenital # m i i (n can al!so be <1(attractive) and ii) complex (absorption or incoherent scattering)

! Index of refraction V (rv ) n(rv ) = 1" E 2" 2 2" 2 V (rv ) = h b$ 3 (r % r ) & h bN nuc # m i m atom number density i ! v V (rv ) = "µv # B mag eff

! Materials with Fe, Ni, Co have Beff s.t. Vmag~Vnuc

r Note: neutron magnetic moment ! v r µ is negative, i.e. “spin up” has Vnuc (r ) µ # Beff v m positive V . n± (r ) = 1" mag E (n can also be i) >1(attractive) ; ii) complex (absorption or incoherent scattering)

! Notes

#2Nb n = 1" 2$ V(r) is generally positive and n<1

! Other interactions r r 1 r r r Vspin"orbit (r , p) = " µ # (p # E ) m

r #(r $) 3 Velectric (r ) = hc"' & d r i ri % r $ !

! Neutron Charge Distribution

Slope gives 2 (<0) 2. Neutron guides

NIST NCNR GUIDE HALL Neutron guides: assume no Bragg scattering; absorption negligible;

n0=1 θ n <1 2" 2 V = h bN m N : number density of atoms b : coherent scattering length

In case of different atoms! i, use the weighted average . b is generally positive (reflection from edge of square well*) so n<1 *see Peshkin &Ringo, Am. J. Phys. 39, 324 (1971) 1 2k 2 2# 2 2 neutrons are totally reflected, if E < V E = mv 2 = h " = h ⊥ " 2 " 2m m$2 " k⊥=ksinθ and λ⊥=λ/sinθ

2" 2 2 mV h < V or sin! " < $ => critical angle θ m#2 2# 2 2 C $ h

58 For Ni, θC/λ = 2.03 mrad/Å (1.73 mrad/Å for natural Ni): m=1 ! ! Basic properties of ideal bent guides All refections are assumed to be specular with reflectivity 1 up to a & mV ) well defined critical angle θ and sin#1 c "C = ( 2 2 %+ with reflectivity 0 above θ . ' 2$ * c h There are two types of reflections: θa> θ a i • Zig-zag reflections (large θa) ! • Garland reflections (never touching the inner wall) (small θa)

γi If the max. reflection angle allows only Garland reflections near the outer wall, then the guide is not efficiently ``filled.’’

a ρ If the filling of the guide will be - θa ≈ θi ρ fairly isotropic (many reflections).

After at least one reflection of all transition from Garland- to neutrons, the angular distribution in the Zig zag reflections guide is well defined. The angles always repeat.

θa- θi The Maier-Leibnitz guide formula

a x x1 x2 ρ - a

θa- θ

θ y ρ

90-θa y = ρ·sin(θa- θ ) ≈ ρ·(θa- θ ); 2 x1 = ρ - ρ·cos(θa- θ ) ≈ ρ/2 ·(θa- θ ) ;

x2 = y ·tanθ ≈ y·θ ≈ ρ (θa- θ ) θ; 2 2 x = x1 + x2 = ρ/2 ·(θa - θ ); θa

2 2 θ = θa - 2x / ρ; The intensity distribution in a long curved guide is a function of λ

To calculate the max. transmitted divergence, we choose: θa = θc = k⊥/ k;

2 ½ Plot of θmax = (θc - 2x/ρ) as function of x for different θc shows added divergence:

For θ = θc the inner wall just does not get touched. In this case the divergence is 0 at the inner wall. θmax

The wavelength λ, corresponding to this θc is 2 called characteristic wavelength λ*. θc > 2a/ρ

2 θc = 2a/ ρ parabolas open θ 2 to the left c < 2a/ρ 0 a x

width of guide Filling factor F (intensity ratio curved guide/straight guide):

2 2 θmax = θa - 2x / ρ

a* a* 1 1 2x 1/2 F dx 1 dx θ *= (2a/ρ) = # " = # $ 2 a"c 0 a 0 %"c a*= γcρ/2 γ

θ θc = 2 c θ* F = 0.93 !

θc θc = θ* F = 2/3 θ c F = 1/6 0 θc =½ θ* a/4 a x

F( = 2 *) = 0.93; F( = */2) = 1/6; F(θc= θ*) = 2/3; θc θ θc θ Multilayer mirrors and supermirrors: & mV ) Can we exceed the critcal angle of “total external reﬂection?” #1 "C = sin ( 2 2 %+ ' 2$ * h Multilayers provide reﬂections from multiple interfaces between different indexes of refraction

n1 n2 ! Incident plane wave; λ0 Reflectivity for normal incidence at one interface 2 ! n1 # n2 " R = $ % & n1 + n2 ' d1 for n1

destructive interference for " = 2n2d2 sin#2

Let n d =n d ! 1 1 2 2 ! Pick n1+n2 small: only a few layers provides high reflectivity for mirror FOR CONSTANT λsinθi

Supermirror: vary nd for quasi-continuous λ and θi single mirror multilayer supermirror

refractive index n < 1 total external reflection

e.g. Ni θc = 0.1 °/Å " = 2ndsin#

! increasing d 1.0 1.0 1.0 0.9 0.9 0.9 0.8 0.8 0.8 y t i

0.7 v 0.7 0.7 i t

0.6 c 0.6 0.6 e l 0.5 f 0.5 0.5 e r reflectivity reflectivity 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 = 5 Å = 5 Å ! = 5 Å 0.0 ! 0.0 λ 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 ｡ " [°] θ [ ] " [°] Multilayer supermirrors: Characterized by m= _θ_m_irror__ θNi 1.73 mrad/Å

From Swiss Neutronics

References: V. F. Turchin, At. Energy 22, 1967. F. Mezei, Novel polarized neutron devices: supermirror and spin component ampliﬁer, Communications on Physics 1, 81, 1976. F. Mezei, P. A. Dagleish, Corrigendum and ﬁrst experimental evidence on neutron supermirrors, Communications on Physics 2, 41, 1977. J. B. Hayter, H. A. Mook, Discrete Thin-Film Multilayer Design for X-Ray and Neutron Supermirrors, J. Appl. Cryst. 22, 35, 19892, 35, 1989. O. Schaerpf, Physica B 156&157 631, 639 (1989) layer sequence λ/4 layer thickness, overlapp of superlattice Bragg peaks 1 J. B. Hayter, H. A. Mook, Discrete Thin-Film Multilayer Design for X-Ray and Neutron Supermirrors, J. Appl. Cryst. 22 (1989) 35

high m - value - more angles m = 2 3 4 5 6 7 8

=> large number of layers, e.g. 350 Ni layer ] [ Ti layer m = 2 ⇒ 120 layers (R ≥ 90%) Å 300

m = 3 ⇒ 400 layers (R ≥ 80%) 250 m = 4 ⇒ 1200 layers (R ≈ 75%)

200 m = 5 ⇒ 2400 layers (R ≈ 63%) thickness 150

layer 100

50 ! = 0.985 m<3.6 1 10 100 1000 10000 number of layers

M. Hino et al., NIMA. 529 (2004) 54

From Swiss Neutronics

m>5 m>1 guides

Replacing a Ni guide (m=1) by a SM guide (m=2) doubles k⊥ and γc. Increasing the guide width from a ⇒ 1.5a increases γ* by 1.5½ ½ ½ and also increases the direct line of sight [Ld = (8aρ) ] by 1.5 .

γ For λ ≠ λ* the intensity m=2 increases by a factor 4. 2γc

γc m=1 New characteristic γ* γc/2 from increase of width 0 a/4 a 1.5a x

For λ ÷ λ* the intensity increases by more than a factor 4 from doubling of γc. [γc/2 ⇒ γc is shown.] Guide losses

Reflectivity:

Garland refl.: lg = 2 ρ γ ; Zig-zag refl.: lz = ρ (γa - γi ); or lz ≈ d/γ; Rn = (1 - Δ)n ≈ 1 - n Δ; for Δ << 1;

for R = 0.97 and L = 1.2 L0; n ≥ 3 (ρ = 2700m; γ = 1/200)

⇒ beam transmission fR by reflectivity R: < fR > = 0.9; higher for long SM-guides!

l = mean length for one reflection from side walls;

n = mean number of reflections; R = reflectivity; L0 = length of free sight; L=100m

2 2 -1/2 Alignment errors: Gauss distrib.: f(h) = exp(-h /hf ) / (π hf ) -1/2 Transmission fa = 1 - L/Lp ⋅hf ⋅π /a; h For L = 1.2 L0 ; a = 3 cm, Lp = 1m, L = 100m and hf = 20 µm:

⇒ mean transmission due to alignment errors: fa = 0.96;

For guide of 20x3 cm, the necessary precision on top / bottom is 7 times worse (0.14 mm). hf = mean alignment error; a = guide width (30 mm); Lp = length of plates; Guide losses

waviness: 1 & 2 # Let the neutron be reflected under angle + instead of . g exp ' γ α γ (')= 2 $ ( 2 ! * ) 'w % 'w " For 6 reflections, L = 1.2 L0 ; k = 0.7k*: fw = 1 - αw /γ*; -4 -3 For αw = 10 , γ* = 1.7 ⋅10 [1Å, Ni]: fw = 0.94; The outer areas of the intensity distribution I(γ) are more affected than the inner ones.

fw : transmission due to waviness; αw = RMS waviness; γ* = characteristic angle of guide

I(γ) without waviness

with waviness

γc γ

-4 αw = 2·10 for the new guides seems acceptable (m=2!)

The angle Δα between the guide sections can be treated as waviness. Δα = 1/27000 for H2. Summary of main guide formulas

-2 -1 ρ = radius of curvature k⊥ [Ni] = 1.07 ·10 Å ↔ m=1; -2 -1 a = width of guide ⇒ k⊥ = 1.07 ·x · 10 Å ; [SM: m=x] x = width from outer surface Supermirrors of n-guides typically have m =2 -2 -1 k⊥ = max. vertical wavevector; k⊥ [glass] = 0.63 ·10 Å ; -2 -1 γ = angular width w.r.t. guide axis k⊥ [Ni-58] = 1.27 ·10 Å ; ½ ½ γc ≈ k⊥ / k = (4π N b) / k = λ (N b/ π)

2 ½ γ = (γc - 2x/ρ) ; γ = angular width at output of curved guide; γc = critical angle of total reflection; ⇒ γ* = (2a/ρ)½; γ* = characteristic angle;

conditions: each neutron is at least once reflected at outer surface and reflectivity is step function up to γc; ½ λ* = 2π (2a/ρ) / k⊥; λ* = characteristic wavelength;

d$ d2$ 1 2 dF Flux dφ/dλ for given source brilliance d φ/dλdΩ and distance z from source: = ! 2 d# source d#d" z

Flux d /d in long straight guide for constant source brilliance d2 /d d 2 φ λ φ λ Ω d$ d $ if angular acceptance in guide γx γy is smaller than angular emittance of source: = !x!y d# d#d" Summary of main guide formulas

1 a* a* 2 1 1 ! 2x " a* = a for γc ≥ γ*; F = dx # = dx%1 $ & 2 F = filling factor of guide a + a + % 2 & a* = ρ γc /2 for γc ≤ γ*; #c 0 0 ) ' ( #c *

2 L0 = 8ρa; L0 = direct line of sight of bent guide;

2 ; Δ= a-(L0/2-dL) /2R Δ = width of direct sight of bent guide; dL = missing length to L0

2 xb= L /(2ρ); xb = lateral deviation from start direction; L = length of guide;

ng = L/(2 ρ γ) ; nz = L/(ρ (γa - γi )); or nz ≈ L γ /a; n = number of reflections; ng for Garland; nz for Zig-zag refl.;

fw = 1 - αw /γ*; fw = loss due to waviness; αw = RMS value of waviness; L = 1.2 ld ;

-1/2 fa = 1 - L/Lp ⋅hf ⋅π /a; fa = loss due to steps; hf = RMS value of alignment error;

2 2 L" ! ) 1 1 & L " ! Loss V in intensity due to gap of length L for a guide of cross scection a×b: V = c ' # $ # c 2 ( a b % 4ab Break Neutron Polarization and Polarimetry

polarizer Spin analyzer Flipper

N0(v) neutrons X P(v) A(v) P T (v) R TA(v) Detector

RExp= Σ( + ) + Δ( - ) =N0T1T2TP [Σ+ΔPR]

Ideal: Γ±=Σ±Δ

__N___+_-_ N__ - __T__+_-_ _T__- Polarizer P= = ρ+-ρ- = N++N- T++T- Flipper: Ru= 1 (unﬂipped); Rf=cosθ≈-1 (ﬂipped)

A A _T___+__-_ T__ Analyzer A = A A T ++ T - Neutron Polarization and Polarimetry

polarizer Spin analyzer Flipper

N0(v) neutrons X P(v) A(v) P T (v) R TA(v) Detector Polarizers: transmit spin-up and down differently Stern Gerlach, magnetic crystals, spin dependent supermirrors; polarized nuclei (scattering or absorption) 1 Figure of merit: #TP 2 " 2

2 P/A Pn (5Å) Tn P T features PSM 99.x% ! 10% 0.1 ﬁxed; limited λ bite 3 He (60%) 80% 30% 0.2 ﬂip P3; P3 varies Polarizing multi-layer & supermirror 1D magnetic layers

b2 = bc2 ± bm;

Plane wave; k 0 non-magn. layers:

b1 = bc1

bc2 + bm b For spin : bc2 + bm c2

bc1

For spin : bc2 – bm ≈ 0 bc2 - bm bc1 • F. Mezei; Commun.Phys.1(1976)81; + Corrigen. : Commun.Phys.2(1977)41; (first paper) • J. Hayter, A Mook: J. Appl. Cryst.: 22(1989)35; (used for supermirror production) Coherent Scattering Lengths b = _A+1____ a A

element b (fm) H -3.74 Be 7.79 C 6.65 Al 3.45 Si 4.15 Ti -3.44 Fe 9.45 Co 2.49 Ni/58Ni 10.3/14.4 Cu/65Cu 7.72/10.6 Cd 4.87-0.7i Nb 2m n = 1" #2 ± #2µB 2$ (2$ )2 h

! Co/Ti polarising supermirrors; K. Andersen, ILL

15 0 2 A)1 10 - ( θ 6 - 5 No magnetic b field substrate B . 0 Nb N -5 Ti Co Ti Co Ti Co air substrate 15 0 2 A)1 10 - ( 6 - 5 B b N(+p) 0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 . 0 Nb N 1 N up N down -5 Ti Co Ti Co Ti Co air 0.8

15 R 0.6 0 2 A)1 10 - ( B 0.4 6 - 2 5 # Nb No θc ! b n 2 =1" >1 0.2 Nb . 0 N(-p) 2$ N 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 -5 ! (°) Ti Co Ti Co Ti Co air ! Fe/Si polarising supermirrors; K. Andersen, ILL

15 0 2 )1 - A 10 6 ( θ λ=3Å : labs = 70cm - 5 b Nb . 0 B N Si substrate -5 Si Fe Si Fe Si Fe S i sub str a te 15 0 2 A)1 10 - ( N(+p) 6 - 5 y b Nb t . 0 i N v -5 i Si Fe Si Fe Si Fe S i sub str a te t 1 c 15 0 e0.8 2 l A)1 10 0.6 - ( f 6 - e0.4 5 No θc ! b Nb N(-p) R 0.2 . 0 N 0 -5 ! Si Fe Si Fe Si Fe S i sub str a te ‘remanent’ polarizing supermirrors - reversible neutorn spin magnetic anisotropy high remanence

1.0

0.5 s M

0 .0 M -0.5

-1.0 -200 -100 0 100 200 H [Oe]

guide field to maintain neutron polarization ≈ 10 G switching of polarizer/analyzer magnetization ⇒ short field pulse ≈ 300 G Crossed Supermirror Kreuz et al./ILL+UM 3He spin ﬁlter: n +3He _> 1H + 3H

polarizer Spin analyzer Flipper

N0(v) neutrons X P(v) A(v) P T (v) R TA(v) Detector

_λ_ σ(J=0)=5333 b; σ(J=1) ≈ 0 (λ0=1.8 Å) λ0 Polarization 0.2 0.4 0.6 0.8 0.9 1.0

0.8

0.6

0.4 Transmission

0.2

0.0 0 2 4 6 8 10 Distance (cm) 3He spin filter Booboo is the Cell

12 cm

GE-180 Glass (boron free)

Cells get “milky” deposit Evidence of Rb depolarization Neutron Polarimetry

polarizer Spin analyzer Flipper

N0(v) neutrons X P(v) A(v) P T (v) R TA(v) Detector λ=5 Å 1.0 1.00 60% 40% 0.8 0.98

0.6 0.96

0.4 0.94 Analyzing Power Analyzing Power 0.2 0.92

0.90 2 4 6 8 5.0 5.5 6.0 6.5 7.0 7.5 8.0 n t! n3t!a 3 a

Opaque Analyzer: For n3tσa large enough (1-A) small enough Can characterize Polarizer AND Flipper Spin Flipper Detector (48 CsI Array) 3He Spin Filter

n+p ___> d + γ (LANSCE & SNS) Pulsed Beam Neutron Polarimetry Flipper

P(v) Experiment A(v) TA/P=T0cosh αA/PPA/P N0(v) FP-12 T (v) R T (v) P A T0=exp(-αA/P- αgl-…)

P/A=tanh αA/PPA/P

M1 M2 M3

M1= N0ε1+B1 M2= N0T1TP ε2+B2

npdgamma M2 Cell Out M2 Polarized normalized to M1 FP12 M2 Unpolarized 2.0 Bragg edges (Al) M2(0)/M2(out) = T0 1.5 M2(P)/M2(0) = cosh αPPP

1.0 (Coulter et al. NIMA 288, 463 (1989) Signal (arb units) 0.5

0.0 2 4 6 Wavelength (Å) Pulsed Beam Neutron Polarimetry Flipper

P(v) Experiment A(v) TA/P=T0cosh αA/PPA/P N0(v) FP-12 T (v) R T (v) P A T0=exp(-αA/P- αgl-…)

P/A=tanh αA/PPA/P

M1 M2 M3

M1= N0ε1+B1 M2= N0T1TP ε2+B2

npdgamma FP12

Offsets/Backgrounds 4

2 Residual (10-3 -3) 0

x10 -2 -4

0.6 expαP

0.4

M2(0)/M2(out) 0.2

2 4 6 wavelength(Å)

-3 2 Residual (10-3) 0

x10 -2 -4

2.5 cosh P _> P =tanh P 2.0 αP P n αP P

1.5 M2(P)/M2(0)

1.0

2 4 6 wavelength (Å) Spin Transport and Spin Flippers dsr r = "sr # B dt r r r ˆ r h ds dB e.g. B = B 0 z , and s = zˆ so = 0 as long as = 0 2 dt dt ! r dB << "B2 is sufficient (adiabatic limit) ! dt ! ! ! A spin in free space is hard to “depolarize” or flip

! Exceptions: “diabatic”, oscillating fields

x x x x x x x x x x x spin up x spin down x x x x

Current sheet spin flipper (also Meissner shield) Resonant Spin Flippers dsr r = "sr # B dt r B = B zˆ + B cos"txˆ 0 1

dsz ! = "B1 dt ! Spin flip: γB1t=π (π pulse)

!npdgamma spin flipper: t=L/v

time after pulse The fast adiabatic spin flipper

Beam area Static gradient field ≈ in z-direction

B > B0 B0 B < B0

P = +1 BRF with fuzzy edges, P = -1 set to ω = ωL = γ⋅B0

B’ = B - ωL/γ = B – B0 ; Spin turn in a frame [‘] rotating with ωL around the z-axis: Bx’ = Brf

B0 z=z’ z=z’ z=z’ z=z’ z=z’

BRF x’ x’ x’ x’ x’

Btotal The fast adiabatic spin flipper (time domain)

d" r r % µ+B# + µ#B+ ( ih = #µ $ B " = #' + µzBz *" dt ( ) & 2 )

B1 ±i" 0t $" +' B± = e " = & ) 2 %" #( B = " t ! z 0

$ B1 i+ 0t ' $ ' µz*0t e $ ' d "+ & 2 ) "+ ih & !) = & )& ) Feynman lectures vol. III dt " B1 #i+ 0t " ! % #!( & e µ * t )% #( % z 0 ( 2

d"+ B1 i$ 0t d"# B1 #i$ 0t ih = µz#0t"+ + e "% ih = e "+ + µz%0t"# dt 2 dt 2 1.0 ! "+ 0.8

0.6 ! ± ! ! 0.4 ! 0.2 "# 0.0 0 200 400 600 800 1000 time ! Rotating frame (wave) picture: cos t= _1_ (eiωt+e-iωt) ω 2

anti-rotating (2w in frame) co-rotating (fixed in frame)

θ B ΔB=B0-ω/γ rot

Bx=B1/2

$ d" ' AFP: Spin follows B adiabatically & << #B1) rot % dt (

Resonant flipper: B=0 Δ! _B_1_ Spin rotates around Bx (Rabi frequency ωR=γ 2 ) From experiment to physics

M1 M2 polarizer Spin analyzer Flipper

N0(v) neutrons X P(v) A(v) TP(v) R TA(v) Detector

N - N ____+______- = C P (1-f) + A N + N n A F false + - background spin ﬂip efﬁcacy analyzing power neutron polarization