by

Roger Pynn

Lectures 10: Polarized Neutrons

Neutron Spin and Magnetic Moment

• Neutron is a fermion with spin ½ & the usual spin operators

• The Pauli spin operators are given by

• Because of its spin, the neutron also has a magnetic moment

• Where γn = -1.913 and μN is the nuclear magneton

• μn is 3 orders of magnitude smaller than the Bohr magneton and antiparallel to the neutron spin

1001

2 ,

00

2 ,

0110

2 ⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡ −=⎥

⎦

⎤⎢⎣

⎡=

hhhzyx s

ii

ss

1-17 J.T 1066.92 −−== xsNnnr

h

rμγμ

srh

r 2=σ

The Neutron Spin Wavefunction & Polarization

• The neutron spin wavefunction is a superposition of “up” and “down”states

• The polarization components of an individual neutron are the expectation values of the appropriate Pauli matrix i.e.

• The polarization P of a neutron beam is the mean value of p in the beam

• Note that:

↓+↑= baχ

22

)*Im(2

)*Re(2

bap

bap

bap

zz

yy

xx

−==

==

==

χσχ

χσχ

χσχ

↓↑

↓↑

+−

=nnnnPz

Behavior of the Neutron’s Spin in a Magnetic Field

• The time evolution of any two-state quantum system can be represented by the movement of a classical vector

where γL is the gyromagnetic ratio of the neutron

• The solutions are

BPdtPd

L

rrr

∧−= γ

1-1-8L T.rad.s 1.832x10 where

)0()()0()sin()0()sin()()0()sin()0()cos()(

−==

=

+=

−=

B

PtPPtPttPPtPttP

L

zz

xLxLy

yLxLx

γω

ωω

ωω

Plug in the Numbers

~29

Turns/m for4 Å neutrons

2918310

N (msec-1)ωL (103 rad.s-1)B(Gauss)

Guiding the Neutron Polarization

• If the direction of a magnetic field varies sufficiently slowly in space, the component of neutron polarization parallel to the applied field is preserved. This is adiabatic polarization rotation.

Drawing from Bob Cywinski

How does a Neutron Spin Behave when the Magnetic Field Changes Direction?

• Distinguish two cases: adiabatic and sudden• Adiabatic – tan(δ) << 1 – large B or small ω – spin and field

remain co-linear – this limit used to “guide” a neutron spin• Sudden – tan(δ) >> 1 – large ω – spin precesses around new

field direction – this limit is used to design spin-turn devices

When H rotates withfrequency ω, H0->H1->H2…, and the spin trajectory is described by a cone rolling on the plane in which H moves

Adiabatic rotation of neutron spin

0

50

100

150

200

250

0 5 10 15 20 25 30 35 40

Hx

Hz

Hx a

nd H

z [Oe]

distance [cm]

φ

0 cm 30 cm

0

20

40

60

80

100

0 5 10 15 20 25 30 35 40

[°]

distance [cm]

φ

x

z

0

2 105

4 105

6 105

8 105

1 106

1.2 106

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

|ωL|/4

|dφ/dt|

0 5 10 15 20 25 30 35

frequ

ency

[rad

/s]

time [ms]

distance [cm]

Note, the relation between distance and time is valid for a neutron with λ = 4 Å.

L

L

dtd

B

ϖφ

γϖ

<<

×−== s-rad/G10833.1 4r

Condition to maintain polarization of the neutron beam.

M. R. Fitzsimmons

Non-Adiabatic Transitions

• If the guide field direction is suddenly changed (i.e. the adiabaticityparameter tan δö¶, the neutron polarization vector will precess about the new field direction.

• If the field is reversed, the neutron polarization is flipped with respect to the field

Graphic courtesy of Bob Cywinski

Viewgraph from Bob Cywinski

Spin Flippers based on Larmor Precession

Sini

B

Send

Sini

BSend

B BBπ/2

BπB B

Direct current spin flippers combine adiabatic and non-adiabatic spin rotation.

Larmor Precession allows the Neutron Spin to be Manipulated using π or π/2 Spin-Turn Coils: Both are Needed for NSE

• The total precession angle of the spin, φ, depends on the time the neutron spends in the B field

v/BdtL γωφ ==

Neutron velocity, v

d

B

][].[].[.135.65

1 turnsofNumber AngstromscmdGaussB λ=

Using Soft Magnetic Films to Rotate Neutron Spins

• A thin film of soft magnetic material is placed in a polarized neutron beam at an angle θ to the beam

• As θ is changed the polarization of the neutron beam changes

220 240 260 280 300 320 340 360-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Data: Data54_polM odel: p i_on_2_fitW eighting: y No w eighting Chi^2/DoF = 0.00092R^2 = 0 .9965 K 2.01933 ±0.00239x0 271.17802 ±0

Pol

ariz

atio

n

R o ta tion S tage A ng le (degrees)

po l10 m icron fo il tuning curve

-1

-0.5

0

0.5

1-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

B

H,σ Bθ neutron

H

-1

-0.5

0

0.5

1-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

Thin Magnetic Films used as π/2 and π Rotators

M

Hg

χneutron

MHg

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

Hg

M

χneutron

MHg

M

χγφsinvMd

=

dv

Rekvedlt

“Wavelength Independent” Flippers can be Made with Permanently Magnetized Films (or Coils)

neutron

B BB1

B2

neutron

B BB1

B2

B

B2

B1

B

B2

B1

1 2 3 4 5 6

-0.2

-0.1

0.1

0.2

C x neutron wavelengthPola

rizat

ion

1 2 3 4 5 6

-0.2

-0.1

0.1

0.2

C x neutron wavelengthPola

rizat

ion

1 2 3 4 5 6

-0.04

-0.02

0.02

0.04

C x neutron wavelength

Neu

tron

Pola

rizat

ion

1 2 3 4 5 6

-0.04

-0.02

0.02

0.04

C x neutron wavelength

Neu

tron

Pola

rizat

ion

2 films

3 films

White Beam Spin Flipper –A rf Gradient-Field Spin Flipper

Hs

H1

Heff

Hs

H1

Hs

H1

HRF=H1cos(ωt) Hs(z) = H0+∇H•z − ω/γ

Viewgraph from M. R. Fitzsimmons

Realization – The ASTERIX rf Flipper

40

60

80

100

120

140

160

-10 0 10 20 30 40

Sta

tic fi

eld,

Hs(z

) [O

e]

Distance along neutron path [cm]

RF field region15 cm

∇H = 3 Oe cm-1

50 feet70 W @ 296kHz

Viewgraph from M. R. Fitzsimmons

Performance of the Asterix rf Flipper

10-7

10-6

10-5

10-4

10-3

10-2

10-1

4 6 8 10 12 14

Polarized beam spectra in 2001

(++)(--)(-+)(+-)

Inte

nsity

[a.u

.]

Wavelength [Å]

proton flash

• (++) = (--) ⇒flippers 99+% efficient for λ > 4Å.

• (++)/(SF) ⇒polarization ~94% for λ > 4Å.

Viewgraph from M. R. Fitzsimmons

Production of Polarized Beams

• Polarizing filters– Usually 3He these days, although pumped protons and rare earths have been

tried– Good for polarizing large, divergent neutron beams– Depend on good polarization of filter material

• Polarizing monochromators– Mainly Heusler alloy (Cu2MnAl) these days– Good when beam monochromatization is also required (e.g. TAS)

• Supermirrors– Very efficient, broad-band polarizers– “cavities” or “benders” are excellent for preparing polarized beams– Disadvantage is that wide angular beam divergence requires devices with

non-uniform transmission

Polarizing Filters

• The polarization and transmission are given by:

2/)( and 2/)( where)cosh( );tanh(

0 −+−+

−

+=−=

=−=

σσσσσσ

σσ σ

p

pNt

p NteTNtP o

Production of polarized 3He at ILL

Making a Polarizing Neutron Monochromator• In a Bravais crystal, the magnetization may be written as a Fourier series:

• This may be combined with the matrix element for nuclear scattering to give an effective scattering length given by:

• Consider a situation in which the magnetization at each lattice site is along z, the quantization direction of the neutron spin, and ┴ to Q

• From the results on the previous slide, the effective scattering lengths are:

and there is no spin flip scattering

• If we can find a FM crystal for which the nuclear and magnetic scattering lengths are equal we can use it to monochromate and polarize neutrons

Factor Form Magnetic theis ).exp()()( where).exp()()( ∑ ∫== ⊥⊥j

jj rdrQirmQFMRQiQFQM

rrrrrrrrrrr

j

Bjj MQF

rbb ⊥+=

rrr.)(

2ˆ 0 σ

μγ

∑j

RQij

jebrr

.

)(2

ˆ :vv For ; )(2

ˆ :uuFor 00 jz

Bjj

jz

Bjj MQFrbbMQFrbb ⊥⊥ −=→+=→

rr

μγ

μγ

• scattering potential is +ve if neutron spinis parallel to i.e. its magnetic moment is anti-parallel to

• thus, if polarized reflection occurs, the neutron spin must be parallel to

Schematic of a Polarizing Monochromator

⊥Mr

Q

u

vUnpolarizedIncidentNeutrons

PolarizedScatteredNeutrons

BSampleMagnetization

x y

zQ

u

⊥Mr⊥M

r

Vertically focussing Heusler alloy monochromator for polarized neutronsA. Freund, R. Pynn, W. G. Stirling and C. M. E. Zeyen; Physica 120B, 86 (1983)

Viewgraph from Bob Cywinski

Neutron Reflection from Magnetic Mirrors

• Since, for a saturated ferromagnet, we can write the magnetic scattering in terms of an effective scattering length, we can deduce a combined nuclear and magnetic SLD for such a material

• We can make a mirror that polarizes a neutron beam by choosing the nuclear and magnetic SLDs to be equal.

• Although the schematic of this process looks much like the polarizing monochromator, the reflection angles are much smaller.

– We also have more control over the mirror than the monochromator because only the average composition matters – we don’t need to rely on nature to hand us the answer

– The first polarizing mirror was demonstrated by in 1951 by Hughes and Burgy

)(1ˆ1 jmagnetic

j

jcoherent

jj bb

volumeb

volume±== ∑∑ρ

Viewgraph from Bob Cywinski

Viewgraph from Bob Cywinski

Polarizing supermirrors

10-5

10-4

10-3

10-2

10-1

100

0 0.02 0.04 0.06 0.08 0.1

R++ Fe/SiR-- Fe/SiR++ 3θ

c SM

R-- 3θc SM

0 1 2 3 4

Pol

ariz

ed n

eutro

n re

flect

ivity

Q [Å-1]⊥

m

0

0.2

0.4

0.6

0.8

1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

0 0.5 1 1.5 2 2.5 3 3.5

Pol

ariz

atio

n (tr

ansm

issi

on)

Q [Å-1]⊥

m

Qmin

= 0.01 Å-1

Qmax

= 0.065 Å-1

Si

100 nm 0.1 nm4

10)(

n

dnd =

F. Mezei and P.A. Dagleish, Comm. on Phys., 2, 41 (1977).

down

up

Partial Cross Sections

IIcohSImag

NSF

z

SImag

SF

z

IIcohSImag

NSF

y

SImag

SF

y

IIcohSImag

NSF

x

SImag

SF

x

dd

dd

dd

dd

dd

dd

dd

dd

dd

dd

dd

dd

dd

dd

dd

dd

dd

dd

dd

dd

dd

dd

dd

dd

⎟⎠⎞

⎜⎝⎛Ω

+⎟⎠⎞

⎜⎝⎛Ω

+⎟⎠⎞

⎜⎝⎛Ω

+⎟⎠⎞

⎜⎝⎛Ω

=⎟⎠⎞

⎜⎝⎛Ω

⎟⎠⎞

⎜⎝⎛Ω

+⎟⎠⎞

⎜⎝⎛Ω

=⎟⎠⎞

⎜⎝⎛Ω

⎟⎠⎞

⎜⎝⎛Ω

+⎟⎠⎞

⎜⎝⎛Ω

+⎟⎠⎞

⎜⎝⎛Ω

+⎟⎠⎞

⎜⎝⎛Ω

+=⎟⎠⎞

⎜⎝⎛Ω

⎟⎠⎞

⎜⎝⎛Ω

+⎟⎠⎞

⎜⎝⎛Ω

+=⎟⎠⎞

⎜⎝⎛Ω

⎟⎠⎞

⎜⎝⎛Ω

+⎟⎠⎞

⎜⎝⎛Ω

+⎟⎠⎞

⎜⎝⎛Ω

+⎟⎠⎞

⎜⎝⎛Ω

+=⎟⎠⎞

⎜⎝⎛Ω

⎟⎠⎞

⎜⎝⎛Ω

+⎟⎠⎞

⎜⎝⎛Ω

+=⎟⎠⎞

⎜⎝⎛Ω

σσσσσ

σσσ

σσσσασ

σσασ

σσσσασ

σσασ

31

21

32

21

31)1(cos

21

32)1(sin

21

31)1(sin

21

32)1(cos

21

2

2

2

2

Blume, Phys. Rev. 130, 1670 (1963); Moon, Riste and Koehler Phys. Rev. 181, 920 (1969

Polarized Neutron Scattering – Various Cases

PG(004) – coherent scatteringIs always NSF

Isotopic incoherent scattering from Niis NSF. Flipper off is NSF.Flipper on is SF

Nuclear spin incoherent scatteringfrom Vanadium is 2/3 SF. Flipperoff is NSF cross section

Paramagnetic scattering from Mn2F. WithP//Q, all mag scattering is SF. With P^Q½ mag scattering is SF and ½ is NSF

Science with Polarized Neutrons

• Most measurements are made with 1-d polarization analysis – Magnetic field is applied to sample and neutron spin component is analyzed in

the field direction– Diffraction

• “Flipping ratio” measurements of form factors, electron spin density distributions usually with single crystals

• 3-directional polarization analysis of diffuse scattering• SANS

– Polarized Neutron Reflectometry (PNR)• Depth dependent vector magnetometry in thin films

– Inelastic scattering• Magnetic excitations and fluctuations

• Generalized polarization analysis is also possible– No magnetic field on sample; incident neutron spin in controlled direction;

analyze neutron spin along any direction• Vector distribution of magnetization in single crystals

Viewgraph courtesy of Bob Cywinski

Viewgraph courtesy of Bob Cywinski

Viewgraph courtesy of Bob Cywinski