Polarization efficiency corr

Polarization efficiency corr

W.C. Chen
3He Neutron Spin Filters

Polarization efficiency correction 

Polarization efficiency correction 

Nomenclature: in generalSM: Supermirror;SF: neutron spin flipper;S: sample;Pn: neutron polarization provided by the polarizer;

P: percentage of neutrons in the “+” state from the polarizer 

An: neutron polarization provided by the analyzer;

A: percentage of neutrons in the “+” state from the analyzer 

εP: the percentage of neutrons flipped by the flipper before th

εA: the percentage of neutrons flipped by the flipper after the 

fP: measured flipping ratio with the flipper before the sample

fA: measured flipping ratio with the flipper after the sample;Here I ignored the spin transport loss and can be added into 

an additional polarizing element.

SM SF S SM

Pn εp Σ++, Σ-+, Σ+-, Σ-- An

SFεA
3He Neutron Spin Filters

Polarization efficiency correction ( Polarization efficiency correction (

Nomenclature: 3He polarizer relatedTn: transmission of unpolarized neutrons through a polarized 

T0: transmission of unpolarized neutrons through a unpolari

TE: transmission of unpolarized neutrons through 3He cell w

T±: transmission if neutron spin is parallel (antiparallel) to 3HF: flipping ratio produced by a polarized 3He cell;

Phe(t): 3He polarization at elapse time t in hours;

PHe0: 3He polarization at time t=0;

T1: 3He polarization relaxation time on the neutron beam lin

l: 3He cell path length;

λ: neutron wavelength;

n: number density of 3He gas;

σ: the 1/v neutron capture cross section given by σ(λ)≈ σ0λ;

SM SF S 3He

Pn εp Σ++, Σ-+, Σ+-, Σ-- T+, T3
3He Neutron Spin Filters

Polarized Polarized 3He related equa

He related equa

Tn = T0 cosh( ) σnlPHeT0 = TEe−σnl

T± = TEe−σnl(1∓PHe )

F = T+

T−

= e

2σnlPHe

PHe(t) = PHe0 exp − t T1 ⎛⎝ ⎜ ⎞ ⎠ ⎟ 

Pn = T + − T−

T+ +T−

= tanh[ ] σ λ( )nHeLPHe

Tn = TE exp( ) −σnHeL cosh( ) σnHeLPHe

= 1− T0

Tn

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2
3He Neutron Spin Filters

Spin transport loss & sample depol

Spin transport loss & sample depol

• Assume spin transport loss and/or sample de

affects the spin-up and spin-down neutrons iand assume the sample depolarization is equtransport loss; 

• It can be shown that the overall transport los

each individual transport loss;

• It can be shown that all spin transport loss an

depolarization could be lumped to either the 
3He Neutron Spin Filters

SM & flipper efficiency ( SM & flipper efficiency (3He analy

He analy

Pn =

I uu Tud+ +Tud− ( )− I ud Tuu+ +Tuu− ( )

I uu Tud+ − Tud− ( )+ I ud Tuu+ − Tuu− ( )

1− 2ε p =

I du Tdd+ +Tdd− ( )− I dd Tdu+ +Tdu− ( )

Pn I dd Tdu+ − Tdu− ( )+ I du Tdd+ − Tdd− ( ( ))

Where Pn = 2P −1

With a 3He analyzer and a NMR based 

Four cross section method to dete

SM and flipper efficiency

whereT: transmission through the polar+(-): neutron spito 3He nuclear sp

U and D: neutroDOWN (D); correspond four 
3He Neutron Spin Filters

Error propagation to SM & flippe

Error propagation to SM & flipp

Four cross section method

( ) δPn 2 = δPn δIuu ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 δIuu ( )2 + δPn δIud ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 δIud ( )2

+

δPn

δTuu+

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 δTuu+ ( )2 + δPn δTuu− ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 δTuu− ( )2 + δPn δTud+ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 δTud+ ( )2 + 

δPn

δI uu

=

2I ud Tud+ Tuu+ − Tud− Tuu− ( )

I uu Tud+ − Tud− ( )+ I ud Tuu+ − Tuu− [ ] ( )2

For Iuu=100 and Iud=10, Tuu+ = Tud+ = 0.2 and Tuu

δPn

δI uu

=1.6%

We can do the same thing for other terms.

δPn

δI ud

=

−2I u

I uu Tud+ − 

( 

[ 
3He Neutron Spin Filters

Error propagation to SM & flippe

Error propagation to SM & flipp

δPn

δTud+

=

−2I uu I uuTud− − I udTuu+ ( )

I uu Tud+ − Tud− ( )+ I ud Tuu+ − Tuu− [ ] ( )2

δPn

δTud−

=

2

I uu T

( 

[ 

δPn

δTuu+

=

2I ud I udTuu− − I uuTud+ ( )

I uu Tud+ − Tud− ( )+ I ud Tuu+ − Tuu− [ ] ( )2

δPn

δTuu−

=

−

I uu T

( 

[ 
3He Neutron Spin Filters

Error propagation to SM & flippe

Error propagation to SM & flipp

Four cross section method

δε ( )p 2 = δε p δPn ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 ( ) δPn 2 + δε p δI du ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 δI du ( )2 + δε p δI dd ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 δI dd ( )

+

δε p

δTdd+

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 δTdd+ ( )2 + δε p δTdd− ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 δTdd− ( )2 + δε p δTdu+ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 δT

( 

δε p

δI dd

=

−2I du Tdu+ Tdd+ −Tdu− Tdd− ( )

Pn I dd Tdu+ −Tdu− ( )+ I du Tdd+ −Tdd− [ ] ( )2

δε p

δI du

=

2I dd Tdu+ Tdd+ −Tdu− Tdd− ( )

Pn I dd Tdu+ −Tdu− ( )+ I du Tdd+ −Tdd− [ ] ( )2

δε p

δPn

=

− 1− 2ε ( p )

Pn
3He Neutron Spin Filters

Error propagation to SM & flippe

Error propagation to SM & flipp

δε p

δTdu+

=

2I dd I ddTdu− − I duTdd+ ( )

Pn I dd Tdu+ −Tdu− ( )+ I du Tdd+ −Tdd− [ ] ( )2

δε p

δTdu−

=

−2I dd I ddTdu+ − I duTdd− ( )

Pn I dd Tdu+ −Tdu− ( )+ I du Tdd+ −Tdd− [ ] ( )2

δε p

δTdd+

=

−2I du I duTdd− − I ddTdu+ ( )

Pn I dd Tdu+ −Tdu− ( )+ I du Tdd+ −Tdd− [ ] ( )2

δε p

δTdd−

=

2I du I ddTdd+ − I duTdu− ( )

Pn I dd Tdu+ −Tdu− ( )+ I du Tdd+ −Tdd− [ ] ( )2
3He Neutron Spin Filters

Flipper efficiency Flipper efficiency vswaveleng

waveleng

0.980.99

1

0 0.1 0.2

Calculated flipper efficiency

δλ/λ

• Mezei 

flipper

• Large 

spread 20%.

• Measu

probabthan 0

• Calcul

probabhigher wavele
3He Neutron Spin Filters

Data reduction Data reduction--polarization efficie

olarization efficie

I ++ = (1+ Pn )Σ++T+ +(1+ Pn )Σ+−T − +(1− Pn )Σ−+T+ +(1− Pn )ΣI −+ = [ ] 1+( ) 1− 2ε Pn Σ++T + + [ ] 1+( ) 1− 2ε Pn Σ+−T − + [ 

1− ( 

1− 2ε 

I +− = (1+ Pn )Σ++T − +(1+ Pn )Σ+−T + +(1− Pn )Σ−+T − +(1− Pn )ΣI −− = [ ] 1+( ) 1− 2ε Pn Σ++T− + [ ] 1+( ) 1− 2ε Pn Σ+−T + + [ 

1− ( 

1− 2ε 

{{

unflipped

flipped

I ++I −+I +−I −−

⎛

⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞⎠ ⎟ ⎟ ⎟ ⎟ ⎟ = T Σ++ Σ−+ Σ+− Σ−− ⎛⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞⎠⎟⎟⎟⎟⎟ Note: T = PaFPn, Pa, F and Pn are 4x4 m

3He analyzer (T+ and T- defined before)

flipper (efficiency of ε) and the SM pol

(efficiency of Pn). Σ++, Σ-+, Σ+-, and Σ-- 

scattering cross sections for the samplesample holder and the cell background. 

SM SF S 3He

Pn ε Σ++, Σ-+, Σ+-, Σ-- T+, T-
3He Neutron Spin Filters

Data reduction Data reduction--polarization efficie

olarization efficie

() () { 

(

( )

(1 ) (1 ) (1 ) (1 )

1 12 1 12 1 12 

(1 ) (1 ) (1 ) (1 )

1 12 1

n n nn

n n 

n n nn

n

I PB T PB T B PT PTI PBT PB T B 

I PB T PB T B PT PTI P B T

ε ε ε

ε

++ ++ + −− − ++ + −−+ ++ + −− − ++ 

+− ++ − −− + ++ − +−− ++ −

=+ +− = + +− ⎡ ⎤ ⎣ ⎦

=+− +−− = +− 

⎡ ⎤ ⎡ ⎤ ⎡ ⎣ ⎦ ⎣ ⎦ ⎣ 

=+ +− = + +− ⎡ ⎤ ⎣ ⎦

=+− +− ⎡ ⎤ ⎣ ⎦ ( ) 1 2 { 

1 12 

(

P BT B 

n 

ε ε

−− + ++ 

⎡ ⎤ ⎡ 

− = +− 

⎣ ⎦ ⎣ 

{{

unflipped

flipped

Note: B=the sample holder and the cell background

SM SF S 3He

Pn ε Σ++, Σ-+, Σ+-, Σ-- T+, TPolarization efficiency corrected backgrou

B++ = I++

(1+ Pn )T+ + (1− Pn )T −

B++ = I−+

[ ] 1+ ( ) 1− 2ε Pn T+ + [ ] 1− ( ) 1− 2ε Pn T −

B++ = 

(1+ Pn 

B++ = 

[ 

1+ ( 

1− 

or or
3He Neutron Spin Filters

Flipping ratio monitorin

Flipping ratio monitorin

3He unflipped (original state)

3He flipped

No sample 

FR = (1+ Pn )T+ + (1− Pn 

[ ] 1+ ( ) 1− 2ε Pn T+ + [ 

1− ( 

1

FR = [1+ (1− 2ε)Pn ]T − + [1− 

(1+ Pn )T − + (1− 
3He Neutron Spin Filters

An interface for polarization efficien

An interface for polarization efficie

• IGOR based comp

software;

• Implemented SM 

efficiency calcula

• Polarization effici

sample not implem
3He Neutron Spin Filters

3He polarization decay online mo

He polarization decay online mo

• online monitoring of 3He polarization d

• Periodically monitoring of flipping ratio

• Periodically monitoring of 3He polariza

measurements.

How T1 uncertainties affect polarization efficie

∂T1

T1

=10% ⇒

∂T1

T1

= 20% ⇒

Transmission 

2.5% in SF and ~2.5% 

∂Transmission

Transmission

= σnlPHe ( )t ∂T1 T1 tT1

5.6% in SF and ~5.6% 
3He Neutron Spin Filters

Online monitoring tools Online monitoring tools --FI

FI

Reflectometry

How T1 uncertainties affect polarization efficienc

∂T1

T1

=10% ⇒

∂T1

T1

= 20% ⇒

Transmission ch

2.5% in SF and ~2.5% in 

∂Transmission

Transmission

= σnlPHe ( )t ∂T1 T1 tT1

5.6% in SF and ~5.6% i
3He Neutron Spin Filters

Online monitoring tools Online monitoring tools --FI

FI

almost ideal case

∂T1

T1

< 2%

FID NMR is a direct way to monitor th

polarization decay. 
3He Neutron Spin Filters

Online monitoring tools Online monitoring tools --FI

FI

Some limitation from FID NMR measureme

not good field homogeneity around the NMR 

restricted space on the instrument for our devi

is typically forced to the location where 

homogeneity is worst for the magnetically sh

What’s limiting our capability for reflec
3He Neutron Spin Filters

Online monitoring tools Online monitoring tools --flipp

flipp

Flipping ratio can also be used to monitor the 3

although it is not straight forward and we do not 

involves in polarization efficiency correction. How

neutron transmission measurements are m

I ++ = (1+ Pn )Σ++T+ +(1+ Pn )Σ+−T − +(1− Pn )Σ−+T+ +(1− Pn )Σ−−I −+ = [ ] 1+( ) 1− 2ε Pn Σ++T + + [ ] 1+( ) 1− 2ε Pn Σ+−T − + [ 

1− ( ) 1− 2ε Pn 

I +− = (1+ Pn )Σ++T − +(1+ Pn )Σ+−T + +(1− Pn )Σ−+T − +(1− Pn )Σ−−I −− = [ ] 1+( ) 1− 2ε Pn Σ++T− + [ ] 1+( ) 1− 2ε Pn Σ+−T + + [ 

1− ( ) 1− 2ε Pn 

{{

unflipped

flipped

SM SF S 3He

Pn ε Σ++, Σ-+, Σ+-, Σ-- T+, T-
3He Neutron Spin Filters

Online monitoring tools Online monitoring tools --flipp

flipp

3He unflipped (original state)

3He flipped

FR = (1+ Pn )T+ + (1− Pn 

[ ] 1+ ( ) 1− 2ε Pn T+ + [ 

1− ( 

1− 

FR = [1+ (1− 2ε)Pn ]T − + [1− (1− 

(1+ Pn )T − + (1− Pn 

PHe ( )t = 1 2σnl ln FR[ 

1+ ( ) 2ε −1 P

(1+ Pn )− FR[ 

1

⎛⎝ ⎜ 

PHe ( )t = 1 2σnl ln FR( ) 1+ Pn − [

[ 

1+ ( ) 2ε −1 Pn 

⎛⎝ ⎜ 

⎨

⎨
3He Neutron Spin Filters

0.620.640.660.68

0.7

0.720.740.760.78

-10 0 10 20 30 40 50 60 70

time, h

3He polarization

y = m1*exp(-m0/m2)

Value Error

m1 0.77759 0.0069146m2 311.48 22.84

Chisq 0.00020637 NA

R 0.99221 NA

Online monitoring tools Online monitoring tools --flipp

flipp

Reflectometry

∂T1

T1

<10%

• Not ide• Use spr

FRs and 

• Can cal

polarizaneed gomeasureknowlepolarizaflipper 

• Better t

very be
3He Neutron Spin Filters

Online monitoring tools Online monitoring tools --FI

FI

SANS

How T1 uncertainties affect polarization efficienc

∂T1

T1

=10% ⇒

∂T1

T1

= 20% ⇒

Transmission ch

2.5% in SF and ~2.5% in 

∂Transmission

Transmission

= σnlPHe ( )t ∂T1 T1 tT1

5.6% in SF and ~5.6% i
3He Neutron Spin Filters

Online monitoring tools Online monitoring tools --FI

FI

Severe limitation from FID NMR measurements becau

homogeneity around the NMR probe due to very rest

instrument. NMR probe is typically forced to the loca

homogeneity is worst for the magnetically shielded sol

sample field also changes the NMR signal. The measure

much shorter than T, making it difficult to monitor 

What’s limiting our capability for SA
3He Neutron Spin Filters

Online monitoring tools Online monitoring tools --flipp

flipp

Limitation and requirement for SA

• Not ideal.• Use spread sheet to match FRs and estimate 

• Can calculate 3He polarization from FR, bu

measurements and good knowledge of SM pefficiency and flipper efficiency.

• Need SM and flipper efficiencies to determi

polarization, hence polarization decay.

• Need a program to communicate the SANS • Different sample depolarization at different 
3He Neutron Spin Filters

3He polarization decay He polarization decay--neutron m

neutron m

Nomenclature: 3He polarizer relatedTn: transmission of unpolarized neutrons through a polarized 

T0: transmission of unpolarized neutrons through a unpolari

TE: transmission of unpolarized neutrons through 3He cell w

T±: transmission if neutron spin is parallel (antiparallel) to 3H

F: flipping ratio produced by a polarized 3He cell;

Phe(t): 3He polarization at elapse time t in hours;

PHe0: 3He polarization at time t=0;

T1: 3He polarization relaxation time on the neutron beam lin

l: 3He cell path length;

λ: neutron wavelength;

n: number density of 3He gas;

σ: the 1/v neutron capture cross section given by σ(λ)≈ σ0λ;

SM SF S 3He

Pn εp Σ++, Σ-+, Σ+-, Σ-- T+, T3
3He Neutron Spin Filters

3He polarization decay He polarization deca

T1 determined from unpolarized neutron tr

Tn = T0 cosh(σnlPHe)T0 = TEe−σnl

PHe(t) = PHe0 exp − t T1 ⎛⎝ ⎜ ⎞ ⎠ ⎟ 

Pn = T + − T−

T+ +T−

= tanh[ ] σ λ( )nHeLPHe

Tn = TE exp( ) −σnHeL cosh( ) σnHeLPHe

= 1− T0

Tn

⎛ ⎝ ⎜ 

PHe ( )t = 1σnl ac

⇒ 

Only at the end of the experime

T0 is measurable.

We need at least three points for this, but they ar

from the user experiment. We are typically able to 

one at the beginning and the other at th
3He Neutron Spin Filters

3He polarization and uncer

He polarization and uncer

T1 = Δt

ln PHei

PHef

⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ 

∂T1

∂PHei

=

−T1

PHei ln PHei

PHef

⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ 

∂T1

∂PHef

=

T1

PHef ln PHei PHef ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ 

( ) ∂T1 2 = ∂T1 ∂PHei ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 ∂PHei ( )2 + 

∂T1

T1

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 = ln PHei PHef ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ −2 ∂PHei PHei ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 + ln PHei PHef ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ −2 ∂PHef PHef ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ 2

T1 determined from unpolarized neutron transm

∂PH

PH

i

P

& 

Method 1
3He Neutron Spin Filters

3He polarization and uncer

He polarization and uncer

T1 determined from unpolarized neutron transm

3He uncertainty propagation

PHe ( )t = PHei exp −t /T1 PHei ,PHef [ ( )]

dPHe ( )t = ∂PHe ∂PHei dPHei + ∂PHe ∂T1 dT1 = ∂PHe ∂PHei dPHei + ∂PHe ∂T1 ∂T1 ∂PH

i 

⎡⎣ ⎢ ⎢ 

=

∂PHe∂PHei +

∂PHe

∂T1

∂T1

∂PHei

⎧⎨ ⎩ ⎫⎬ ⎭ dPHei + ∂PHe ∂T1 ∂T1 ∂PHef d

( ) ∂PHe 2 = ∂PHe ∂PHei + ∂PHe ∂T1 ∂T1 ∂PHei ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ 2 ∂PHei ( )2 + ∂PHe ∂T1 ∂T1 ∂PHef ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ 2 

⇒

∂PHe

PHe

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 = 1− tΔt ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 ∂PHei PHei ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 + tΔt ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 ∂PHef PHef ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ 2
3He Neutron Spin Filters

3He polarization decay He polarization deca

T1 = Δt

ln PHei

PHef

⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ 

T1 determined from unpolarized neutron transm

PHe ( )t = PHei exp(−t / Δt

ln PHei

PHef

⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ 

⎛

⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞⎠ ⎟ ⎟ ⎟ ⎟ ⎟ = PH

∂PHe

PHe

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 = 1− tΔt ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 ∂PHei PHei ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 + tΔt ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 ∂PHef PHef ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ 

Method 2
3He Neutron Spin Filters

3He polarization uncertainty co

He polarization uncertainty co

∂PHe

PHe

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 = 1− tΔt ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 ∂PHei PHei ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 + tΔ

⎛ ⎝ ⎜ 

Method 1Method 2

∂PH

PH

i

P

& 

∂PHe

PHe

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 = 1− tΔt ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 ∂PHei PHei ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 + tΔt ⎛ ⎝ ⎜ 

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 10 20 30 40 50 60

error using T1method 1 or Method 2

PHe (t) uncertainty (fractional)

time, h

∂T1

T1

≈ 15%