Neutron Optics and Polarization - NIST · PDF fileNeutron Optics and Polarization ... Neutron...
Transcript of Neutron Optics and Polarization - NIST · PDF fileNeutron Optics and Polarization ... Neutron...
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