Neutron Optics and PolarizationT. 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 1989Rauch and Werner, Neutron Interferometry, Oxford 2000Fermi: Nuclear Physics (notes by Orear et al. U. Chicago Press - 1949)QM Text (e.g. Griffiths)

OpticsOptics: 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)

!

" >> a (atomic spacing)Useful when

deBroglie: massive particles behave as waves

!

k =2"

#=p

h

mc2 = = 939.6 MeV

= 197.3 MeV-fm = 1973 eV-Å

v = 2200 m/s λ = 1.8 Å(thermal neutrons - 300° K)

Note also:

!

hc

!

" #1

T

!

"2 =4# 2

(hc)2

2mc2kBT

!

kBT =p2c2

2mc2

Wave Properties• Polarization

• Reflection

• Refraction

• Interference Superposition

• Diffraction

mirrorir (angle of incidence = angle of reflection)

it

ni sin i = nt sin t

+ =

m λ=W sin θ

n=c0/cc0=2.97x108 m/s

xEy

Vertically polarized lightdifferent meaning

The wave equations for light and matter waves in vacuum:

Time dependent:

Helmholtz equation:

EM wave equation (E, B)

!

"2#$

1

c2

%2#

%t2

= 0

!

"2#+ 2i

m

h

$#

$t= 0

Schrödinger equation

!

"2#(

v r ) + k

2#(

v r ) = 0

!

k =2"

#=p

h

!

"(r r ,t) = a

ke

i(r k #

r r $%

kt )

!

k2

=2mE

h2

Time independentSchroedinger equation

!

k2

=E2

(hc)2Dispersion relations:

!

vph ="

k= c

Phase velocity:

!

vph ="

k1+

m2c2

p2

#"

k

c

v

!

(E = h")

!

E = p2c2 + m2

c4( )

a

Interactions V(r)

!

"2#(

v r ) +

2m

h2

E $V (v r )[ ]#(r) = 0 Time independent

Schroedinger equation

a=- __: scattering length(-5 fm < a < 15 fm)

ka=-δ0 ~ 10-4

!

"(v r ) = e

ikz+

f (#)

re

iv k $

v r Incoming plane wave

Outgoing spherical wave

!

f (") =1

2ik(2l +1)[e

2i# l $1]Pl (cos")l

%

!

f (") =1

2ik[e2i# 0 $1] =

1

[k cot#0$ ik]

Partial waves

s-wave scattering

!

f (") = #a + ika2

+O(k2)

!

"0

= #kro

V(r)=0 at ro

!

f (") # $a

δ0k

Coherent Scattering Lengthsb = _____ aA+1

A

4.87-0.7iCd7.72/10.6Cu/65Cu10.3/14.4Ni/58Ni

2.49Co 9.45Fe-3.44Ti 4.15Si 3.45Al 6.65C 7.79Be-3.74Hb (fm)element

!

f (") =1

2ik[e2i# 0 $1]

!

"0

= #ka

Index of refraction

!

"2#(

v r ) +

2m

h2

E $V (v r )[ ]#(r) = 0 Time independent

Schroedinger equation

!

"2#(

v r ) + K

2#(

v r ) = 0

!

K2 =2m

h2

E "V (v r )[ ]

For light:

!

c"c

n=K

k

!

n(v r ) = 1"

V (v r )

E

In general, n is a tensor, i.e. V(r) depends on propogation direction

(n can also be <1(attractive) and ii) complex (absorption or incoherent scattering)

!

V (v r ) =

2"h2

mi

# b$ 3(r r %

r r

i) Fermi Pseudopotenital

Index of refraction

!

n(v r ) = 1"

V (v r )

E

(n can also be i) >1(attractive) ; ii) complex (absorption or incoherent scattering)

!

Vnuc(v r ) =

2"h2

mi

# b$ 3(r r %

r r i) &2"h

2

mbN

!

Vmag (v r ) = "

v µ #

v B eff

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

!

n±(v r ) = 1"

Vnuc (v r ) m

r µ #

r B eff

E

atom number density

Note: neutron magnetic momentµ is negative, i.e. “spin up” has positive Vmag.

Notes

!

n = 1"#2Nb

2$

V(r) is generally positive and n<1

Other interactions

!

Vspin"orbit (r r ,

r p ) = "

1

m

r µ # (

r p #

r E )

!

Velectric

(r r ) = hc"

#( $ r )

ri% $ r

&i

' d3r

Neutron Charge Distribution

Slope gives<rq

2> (<0)

2. Neutron guides

NIST NCNR GUIDE HALL

Neutron guides: assume no Bragg scattering; absorption negligible;

N : number density of atoms

b : coherent scattering length

In case of different atoms i, use the weighted average <ni·bc>.

b is generally positive (reflection from edge of square well*) so n<1*see Peshkin &Ringo, Am. J. Phys. 39, 324 (1971)

neutrons are totally reflected, if E⊥ < V

!

V =2"h

2

mbN

n0=1n <1

!

E" =1

2mv"

2=

h2k"2

2m=2# 2

h2

m$"2

θ

k⊥=ksinθ and λ⊥=λ/sinθ

!

2" 2h2

m#$2

<V

!

sin" <mV

2# 2h2$or

For 58Ni, θC/λ = 2.03 mrad/Å (1.73 mrad/Å for natural Ni): m=1

k

=> critical angle θC

There are two types of reflections:

• Zig-zag reflections (large θa)

• Garland reflections (never touching theinner wall) (small θa)

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

If θa ≈ θi the filling of the guide will befairly isotropic (many reflections).

Basic properties of ideal bent guides

transition from Garland- toZig zag reflections

γi

ρ

ρ-a

θa- θi

θa> θi

!

"C

= sin#1 mV

2$ 2h2%

&

' (

)

* +

a

All refections are assumed to bespecular with reflectivity 1 up to awell defined critical angle θc andwith reflectivity 0 above θc .

After at least one reflection of allneutrons, the angular distribution in theguide is well defined. The anglesalways repeat.

y = ρ·sin(θa- θ ) ≈ ρ·(θa- θ );

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

x2 = y ·tanθ ≈ y·θ ≈ ρ (θa- θ ) θ;

x = x1 + x2 = ρ/2 ·(θa2- θ2);

θ 2 = θa 2 - 2x / ρ;

ρ

ρ - a

θa

θ

xa

θa- θ

x2x1

y

The Maier-Leibnitz guide formula

90-θa

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;

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

θmax

For θ = θc the inner wall just does not get touched.

In this case the divergence is 0 at the inner wall.

The wavelength λ, corresponding to this θc is called characteristic wavelength λ*.

xa0

θc2 = 2a/ρ

width of guide

θc2 > 2a/ρ

θc2 < 2a/ρ

parabolas open to the left

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

F(θc= θ*) = 2/3; F(θc= 2θ*) = 0.93; F(θc= θ*/2) = 1/6;

xa0

θc = θ*

γθc = 2θ*

θc =½ θ*a/4

F = 0.93

F = 2/3

F = 1/6

θc

θc

θc

θmax2 = θa 2 - 2x / ρ

θ *= (2a/ρ)1/2

a*= γcρ/2

!

F =1

a"c

"dx0

a*

# =1

a1$

2x

%"c

2dx

0

a*

#

Multilayer mirrors and supermirrors:Can we exceed the critcal angle of “total external reflection?”

!

"C

= sin#1 mV

2$ 2h2%

&

' (

)

* +

n1 n2

Incident plane wave; λ0Reflectivity for normal incidence

at one interface2

1 2

1 2

n nR

n n

! "#= $ %

+& '

constructive interference fordestructive interference for

d1

Multilayers provide reflections from multiple interfaces between different indexes of refraction

Let n1d1=n2d2Pick n1+n2 small: only a few layers provides high reflectivity for mirror

FOR CONSTANT λsinθi

Supermirror: vary nd for quasi-continuous λ and θi

!

" = 2n1d1sin#

1

!

" = 2n2d2sin#

2

for n1<n2

refractive index n < 1

total external reflection e.g. Ni θc = 0.1 °/Å

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

! = 5 Å

refle

ctiv

ity

" [°]0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.00.10.20.30.40.50.60.70.80.91.0

λ = 5 Å

refle

ctiv

ity

θ [｡]

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

! = 5 Å

refle

ctiv

ity

" [°]

!

" = 2nd sin#

single mirror multilayer supermirror

increasing d

Multilayer supermirrors: Characterized by m= _____θmirrorθNi

References:V. F. Turchin, At. Energy 22, 1967.F. Mezei, Novel polarized neutron devices: supermirror and spin component amplifier, Communications onPhysics 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)

From Swiss Neutronics

1.73 mrad/Å

layer sequence λ/4 layer thickness, overlapp of superlattice Bragg peaks1 J. B. Hayter, H. A. Mook, Discrete Thin-Film Multilayer Design for X-Ray and Neutron Supermirrors, J. Appl. Cryst. 22 (1989) 35

1 10 100 1000 10000

50

100

150

200

250

300

350 Ni layer Ti layer

876543

! = 0.985

laye

r th

ickn

ess

[Å]

number of layers

m = 2 high m - value - more angles

=> large number of layers, e.g. m = 2 ⇒ 120 layers (R ≥ 90%) m = 3 ⇒ 400 layers (R ≥ 80%) m = 4 ⇒ 1200 layers (R ≈ 75%) m = 5 ⇒ 2400 layers (R ≈ 63%)

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

m<3.6

m>5

From Swiss Neutronics

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

and also increases the direct line of sight [Ld = (8aρ)½] by 1.5½.

xa0

m=1

γ

m=2

a/4

New characteristic γ*from increase of width

γc

γc/2

2γc

1.5a

For λ ≠ λ* the intensity increases by a factor 4.

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

m>1 guides

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

Alignment errors: Gauss distrib.: f(h) = exp(-h2/hf2) / (π-1/2 hf )

Transmission fa = 1 - L/Lp ⋅hf ⋅π-1/2 /a;

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;

h

waviness:

Let the neutron be reflected under angle γ + α instead of γ.

For 6 reflections, L = 1.2 L0 ; k = 0.7k*: fw = 1 - αw /γ*;For αw = 10-4, γ* = 1.7 ⋅10-3 [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

( ) !!"

#$$%

&

''(

')*=' 2

w

2

2w

exp1

g

I(γ)

γγc

with waviness

without waviness

αw = 2·10-4 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.

Guide losses

Summary of main guide formulas

γ = (γc 2 - 2x/ρ)½; 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;

ρ = radius of curvaturea = width of guidex = width from outer surfacek⊥ = max. vertical wavevector;γ = angular width w.r.t. guide axisγc ≈ k⊥ / k = (4π N b)½ / k = λ (N b/ π)½

λ* = 2π (2a/ρ)½ / k⊥; λ* = characteristic wavelength;

k⊥ [Ni] = 1.07 ·10-2 Å-1 ↔ m=1; ⇒ k⊥ = 1.07 ·x · 10-2 Å-1; [SM: m=x]Supermirrors of n-guides typically have m =2k⊥ [glass] = 0.63 ·10-2 Å-1;k⊥ [Ni-58] = 1.27 ·10-2 Å-1;

dF

z

1

dd

d

d

d

2source

2

!"#

$=

#

$Flux dφ/dλ for given source brilliance d2φ/dλdΩ and distance z from source:

yx

2

dd

d

d

d!!

"#

$=

#

$Flux dφ/dλ in long straight guide for constant source brilliance d2φ/dλdΩif angular acceptance in guide γx γy is smaller than angular emittance of source:

Summary of main guide formulas

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.;

L02= 8ρa; L0 = direct line of sight of bent guide;

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

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

xb= L2/(2ρ); xb = lateral deviation from start direction; L = length of guide;

F = filling factor of guide

1a* a* 2

2

c c0 0

1 1 2xF dx dx 1

a a

! "= # = $% &% &# ' ( #) *

+ +a* = a for γc ≥ γ*;a* = ρ γc 2/2 for γc ≤ γ*;

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

ab4

L

b

1

a

1

2

LV

2c

2c !"

#$%

&'(

)#

!"=Loss V in intensity due to gap of length L for a guide of cross scection a×b:

Break

Neutron Polarization and Polarimetry

neutronsP(v)TP(v)

SpinFlipper

N0(v)

RA(v)TA(v)

X

Detector

polarizer analyzer

Flipper: Ru= 1 (unflipped); Rf=cosθ≈-1 (flipped)

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

Ideal: Γ±=Σ±Δ

P= _________ = ρ+-ρ- = _________N+- N-N++N-

T+- T-T++T-

Polarizer

A = _________TA+- TA

TA++ TA

-Analyzer

Neutron Polarization and Polarimetry

P/A Pn (5Å) Tn P2T features PSM 99.x% 10% 0.1 fixed; limited λ bite3He (60%) 80% 30% 0.2 flip P3; P3 varies

Polarizers: transmit spin-up and down differently

Figure of merit:

!

1

" 2#TP

2

Stern Gerlach, magnetic crystals, spin dependent supermirrors;polarized nuclei (scattering or absorption)

neutronsP(v)TP(v)

SpinFlipper

N0(v)

RA(v)TA(v)

X

Detector

polarizer analyzer

For spin : bc2 + bm

For spin : bc2 – bm ≈ 0

bc2

bc2 + bm

bc1

bc2 - bmbc1

magnetic layers b2 = bc2 ± bm;

non-magn. layers: b1 = bc1

Plane wave; k0

1DPolarizing multi-layer & supermirror

• 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 Lengthsb = _____ aA+1

A

4.87-0.7iCd7.72/10.6Cu/65Cu10.3/14.4Ni/58Ni

2.49Co 9.45Fe-3.44Ti 4.15Si 3.45Al 6.65C 7.79Be-3.74Hb (fm)element

!

n = 1" #2Nb

2$± #2µB

2m

(2$h)2

Co/Ti polarising supermirrors; K. Andersen, ILL

No θc !

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

N upN down

0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2

R

! (°)

!

n2

=1"#2Nb

2$>1

No magnetic field

B

B

-5

0

5

10

15

Ti Co Ti Co Ti Co air

N.b (10

-6 A-

2 )

Nb

-5

0

5

10

15

Ti Co Ti Co Ti Co air

N.b (10

-6 A-

2 )

Nb

N(+p)

-5

0

5

10

15

Ti Co Ti Co Ti Co air

N.b (10

-6 A-

2 )

Nb N(-p)

substrate B

substrate

θ

Fe/Si polarising supermirrors; K. Andersen, ILL

-5

0

5

10

15

Si Fe Si Fe Si Fe Si substrate

N.b (10

-6 A-

2 )

Nb

-5

0

5

10

15

Si Fe Si Fe Si Fe Si substrate

N.b (10

-6 A-

2 )

N(-p)Nb

-5

0

5

10

15

Si Fe Si Fe Si Fe Si substrate

N.b (10

-6 A-

2 )

Nb

N(+p)

0

0.2

0.4

0.6

0.8

1

Reflectivity

!

No θc !

Si substrateB

θ λ=3Å : labs = 70cm

-200 -100 0 100 200-1.0

-0.5

0.0

0.5

1.0

M / Ms

H [Oe]

‘remanent’ polarizing supermirrors - reversible neutorn spin

magnetic anisotropy

high remanence

guide field to maintain neutron polarization ≈ 10 Gswitching of polarizer/analyzer magnetization ⇒ short field pulse ≈ 300 G

Crossed SupermirrorKreuz et al./ILL+UM

neutronsP(v)TP(v)

SpinFlipper

N0(v)

RA(v)TA(v)

X

Detector

polarizer analyzer

3He spin filter: n +3He _> 1H + 3H

σ(J=0)=5333 __ b; σ(J=1) ≈ 0λλ0

(λ0=1.8 Å)

1.0

0.8

0.6

0.4

0.2

0.0

Tra

nsm

issio

n

1086420

Distance (cm)

0.2 0.4 0.6 0.8 0.9

Polarization

Booboo is the Cell

GE-180 Glass(boron free)

12 cm

3He spin filter

Cells get “milky” depositEvidence of Rb depolarization

Neutron Polarimetry

1.0

0.8

0.6

0.4

0.2

Analy

zin

g P

ow

er

8642n3t!a

1.00

0.98

0.96

0.94

0.92

0.90

Analy

zin

g P

ow

er

8.07.57.06.56.05.55.0n3t!a

λ=5 Å

Opaque Analyzer: For n3tσa large enough (1-A) small enoughCan characterize Polarizer AND Flipper

40%

60%

neutronsP(v)TP(v)

SpinFlipper

N0(v)

RA(v)TA(v)

X

Detector

polarizer analyzer

3He Spin FilterSpin Flipper Detector (48 CsI Array)

n+p ___> d + γ (LANSCE & SNS)

Flipper

FP-12P(v)TP(v)

M3

N0(v) RA(v)TA(v)

M2M1

Experiment

M1= N0ε1+B1 M2= N0T1TP ε2+B2

M2(0)/M2(out) = T0 M2(P)/M2(0) = cosh αPPP

(Coulter et al. NIMA 288, 463 (1989)

TA/P=T0cosh αA/PPA/P

T0=exp(-αA/P- αgl-…)P/A=tanh αA/PPA/P

2.0

1.5

1.0

0.5

0.0

Sig

na

l (a

rb u

nits

)

642Wavelength (Å)

M2 Cell Out M2 Polarized M2 Unpolarized

normalized to M1Bragg edges (Al)

Pulsed Beam Neutron Polarimetry

npdgammaFP12

Flipper

FP-12P(v)TP(v)

M3

N0(v) RA(v)TA(v)

M2M1

Experiment

M1= N0ε1+B1 M2= N0T1TP ε2+B2

TA/P=T0cosh αA/PPA/P

T0=exp(-αA/P- αgl-…)P/A=tanh αA/PPA/P

Pulsed Beam Neutron Polarimetry

npdgammaFP12

Offsets/Backgrounds

0.6

0.4

0.2

M2(0

)/M

2(o

ut)

642

wavelength(Å)

-4-2024

x10

-3

2.5

2.0

1.5

1.0

M2

(P)/

M2

(0)

642

wavelength (Å)

-4-2024

x10

-3

Residual (10-3)

Residual (10-3)

coshαPPP _> Pn=tanhαPPP

expαP

Spin Transport and Spin Flippers

!

dr s

dt= "

r s #

r B

e.g. , and so as long as

!

r s =

h

2

ˆ z

!

r B = B

0ˆ z

!

dr s

dt= 0

!

d

r B

dt= 0

!

d

r B

dt<< "B2 is sufficient (adiabatic limit)

A spin in free space is hard to “depolarize” or flip

Exceptions: “diabatic”, oscillating fieldsxxxxxxxxxxxxxxxx

spin up spin down

Current sheet spin flipper (also Meissner shield)

Resonant Spin Flippers

!

dr s

dt= "

r s #

r B

!

r B = B

0ˆ z + B

1cos"tˆ x

!

dsz

dt= "B

1

Spin flip: γB1t=π (π pulse)

npdgamma spin flipper: t=L/v

time after pulse

B1

The fast adiabatic spin flipper

Static gradient field ≈ in z-directionBeam area

BRF with fuzzy edges,set to ω = ωL = γ⋅B0

B0B > B0 B < B0

P = +1 P = -1

Spin turn in a frame [‘] rotating with ωL around the z-axis:

z=z’

x’

z=z’

x’

z=z’

x’

z=z’

x’

z=z’

x’

B0

BRF

Btotal

Bx’ = Brf

B’ = B - ωL/γ = B – B0 ;

Feynman lectures vol. III

The fast adiabatic spin flipper (time domain)

!

ihd"

dt= #

r µ $

r B ( )" = #

µ+B# + µ#B+

2+ µ

zB

z

%

& '

(

) * "

!

" ="+

"#

$

% &

'

( )

!

B± =B1

2e

±i"0t

!

Bz

= "0t

!

ihd

dt

"+

"#

$

% &

'

( ) =

µz*0t

B1

2ei+

0t

B1

2e# i+

0t µ

z*0t

$

%

& & &

'

(

) ) )

"+

"#

$

% &

'

( )

!

ihd"+

dt= µ

z#0t"+ +

B1

2ei$

0t"%

!

ihd"#

dt=B1

2e#i$

0t"+ + µ

z%0t"#

1.0

0.8

0.6

0.4

0.2

0.0

!±

10008006004002000

time

!

"#

!

"+

Rotating frame (wave) picture: cosωt= __ (eiωt+e-iωt)12

ΔB=B0-ω/γ

Bx=B1/2

Brot

AFP: Spin follows Brot adiabatically

θ

Resonant flipper: ΔB=0Spin rotates around Bx (Rabi frequency ωR=γ ___)

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

!

d"

dt<< #B

1

$

% &

'

( )

B12

From experiment to physics

neutronsP(v)TP(v)

SpinFlipper

M2

N0(v)

RA(v)TA(v)

M1

X

Detector

polarizer analyzer

__________ = C Pn A F (1-f) + Afalse N+ - N-N+ + N- background

spin flip efficacy analyzing powerneutron polarization