will be presented Outline - Oxford NeutronSchool · B. Farago Polarized Neutrons BNC Neutron school Isotop and spin incoherence 4 bi can have a distribution because: - different isotopes

B. Farago Polarized Neutrons

BNC Neutron school

Outline

1

Outline

•Basics •Magnetic scattering

•Spin manipulation •Instruments

This presentation might differ from the one which really will be presented


BNC Neutron school

Basics

2

Basics


BNC Neutron school 3

Incoherent scattering

Reminder: The scattering cross section:

dsdW

= Âi, j

bib j exp

⇣i~q

⇣~Ri�~Rj

⌘⌘

Suppose that at position Ri we can have different scattering length with a certain probability distribution

hbib ji = hbiihb ji+di j

⇣⌦b2

i↵�hbii2

⌘

dsdW

= Âi, jhbiihb jiexp

⇣i~q

⇣~Ri�~Rj

⌘⌘+Â

i

⇣⌦b2

i↵�hbii2

⌘

I

coherent

(~q)+ I

incoherent

(~q)


BNC Neutron school

Isotop and spin incoherence

4

bi can have a distribution because:

- different isotopes exist

- the nucleus has a spin I => with the neutron 1/2 spin it forms two possible states

the nuclear spin is usually randomly oriented EXCEPT very

low T or very high B

I+1/2 => 2(I+1/2)+1 states with scattering length b+ I-1/2 => 2(I-1/2)+1 states with scattering length b-

Spin Incoherence

Isotope Incoherence

hbii = b+I +1

2I +1+b�

I2I +1

Coherent

Isotope Incoherence

⇣⌦b2

i↵�hbii2

⌘= (b+�b�)2 I(I +1)

(2I +1)2

Incoherent


BNC Neutron school

Polarized beam scattering

5

σx and σy changes the spin state of the neutron => spin flip scattering σz does not => non spin flip scattering

If the nuclear spin I is randomly oriented in space each one has 1/3 probability thus: Spin Incoherent scattering 2/3 spin flip 1/3 non spin flip P => -1/3P

Let’s define the polarization of the beam as: ~P = 2h~si = h~siIn terms of Pauli matrices

sx

=✓

0110

◆sy =

✓0�ii 0

◆sz =

✓1 00�1

◆

Scattering length on a nucleus with spin I:

b = A+12

BsI

A = b+I +1

2I +1+b�

I2I +1

B =2(b+�b�)

2I +1


BNC Neutron school

Magnetic scattering

6

Magnetic scattering


BNC Neutron school

Magnetic Interactions 1

7

The neutron has a 1/2 spin => magnetic moment µn= γn µN

µN nuclear Bohr magneton and γn = -1.913

With a magnetic field the interaction potential Lovesey 1986:

V (~R) =�~µn

~B = g

n

µ

N

h2µ

B

curl

⇣~s⇥~

R

|R3|

⌘� e

2m

e

c

⇣~p

e

~s⇥~R

|R3| +~s⇥~R

|R3| ~pe

⌘i

~s = eletron spin operator,~pe

= eletron momentum operator

~s = neutron spin operator


BNC Neutron school

Magnetic Interactions 2

8

The matrix element (scattering probability) becomes:

or in terms of the magnetization and changing to integral to account for the spatial extent of the electrons

Of the sample spins (magnetization) ONLY THE COMPONENT PERPENDICULAR to q contributes !!This

is fundamentally different from the nuclear spin!

hk0|VM |ki = �r0sQ? with r0 =gne2

mec2

ˆQ? = Âi

exp(i~q~ri)✓

q⇥ (~s⇥ q)� ih |~q| (q⇥~pi)

◆with q =

~q|~q|

~Q =� 1

2µB

Zd~r exp

⇣i~k~r

⌘~M (~r) and ~Q? = ~Q� q

⇣~Qq

⌘


BNC Neutron school

Interference term

9

The scattered intensity is the Fourier transform of the self correlation function of the scattering length density

Alternatively if the Fourier transform of the scattering length density is F(q)

S(q) = F(q)F*(q)

If F(q) = FN(q)+FM(q) in the most generic case there will be four terms: • nuclear

• magnetic

• nuclear magnetic interference

• chiral


BNC Neutron school

D20

10

3000

2000

1000

0

coun

ts /

5 m

in

140120100806040202Theta/deg

369.5Å^3

369.0

368.5

368.0

367.5

367.0

366.5

366.0

365.5

unit

cel

l vol

ume

250K200150100500sample temperature

5

4

3

2

1

0

Bohr magnetons

Unpolarized neutrons - comparable intensity to

nuclear - identified by a priory

knowledge - temperature dependence


BNC Neutron school

3D Polanal X

11

neutron kinneutron kout Q scattering vector

neutron beam polarization direction

50% non spin flip scattering

50% spin flip scattering


BNC Neutron school

3D Polanal Y

12



spin flip scattering

spin flip scattering


BNC Neutron school

3D Polanal Z

13






BNC Neutron school

3D pol anal coils

14


BNC Neutron school

3D pol anal simple

15

UPX = N +12

M +13

I

DOWNX =12

M +23

I

UPZ = N +12

M +13

I

DOWNZ =12

M +23

I

UPY = N +13

I

DOWNY = M +23

I

N the nuclear scattering M the magnetic scattering I the incoherent scattering


BNC Neutron school

Inelastc corrections

16

neutron kin




Q scattering vectorneutron kout

When the scattering is non negligibly quasielastic

with distribution

not quite !


BNC Neutron school

3D pol anal complicated

17

UPX = N1+ f g

2+

12

M� g f2

M sin2 e+3�g f

6I

DOWNX = N1�g

2+

12

M +g2

M sin2 e+3+g

6I

UPZ = N1+ f g

2+

12

M +3�g f

6I

DOWNZ = N1�g

2+

12

M +3+g

6I

UPY = N1+ f g

2+

1�g f2

M +g f2

M sin2 e+3�g f

6I

DOWNY = N1�g

2+

1+g2

M� g2

M sin2 e+3+g

6I

g is the polarizing efficiency f the flipper efficiency N the nuclear scattering M the magnetic scattering I the incoherent scattering ε is the angle of Q to Qelastic


BNC Neutron school

Spin manipulation

18

Spin manipulation


BNC Neutron school

Larmor precession

19

|ci= a |+i+b |�i where a and b can be complex and p

a2 +b2 = 1

and conveniently |ci= e�ij/

2

cos

q2

|+i+ eij/

2

sin

q2

|�i

Time evolution of the neutron spin in magnetic field

∂s∂t

=1h[s,Hs] =�

g2

hs,

⇣s~H

⌘i= ... =�g

⇣~H⇥ s

⌘

The Polarization of the beam P=<σ> behaves as a “classical” Larmor precession

γL =2957 Hz/Gauss DfDx

⇥deg

�cm

⇤= 2.65l

⇥A⇤

H [Gauss]

Quantum mechanical description of the neutron spin state:

this leads to h ˆsx

i = sinqcosj, h ˆsy

i = sinqsinj, h ˆsx

i = cosq


BNC Neutron school

Adiabaticity

20

What happens if the direction of the magnetic field changes in space?

The moving neutron will see a B field which changes its direction in time

Two limiting cases

if ωB << ωL it will follow adiabatically

if ωB >> ωL it will start to precess around the new field direction

neutron trajectory

magnetic field directionadiabatic example:


BNC Neutron school

Mezei flipper

21

Non adiabatic example : Mezei flipper


BNC Neutron school

Real flippers

22


BNC Neutron school

RF flipper

23

RF flipper

Brf oscillating ωRF radiofrequency field

P0 beam polarizationB0 static field

Makes up a π flipper • if B0γL = ωRF

• and BRF is just enough for a π turn during the flight time


BNC Neutron school

Haussler

24

Haussler Xtal Cu2MnAl (111):

• FM(q) = FN(q) for one of the Bragg reflections • easy to saturate • grow single crystal • controlled mosaicity • low λ/2 contamination (or filter)


BNC Neutron school

Multilayer

25

} d

λ

θ®

® ΔQ

0

0.2

0.4

0.6

0.8

1

1.2

0 0.02 0.04 0.06 0.08 0.1

Ref

lect

ivity

Q (Å-1 )

Δλλ

= 2d2 (f

1- f

2)

π

4N2d4(f1- f

2)2

π2n4R =

θ ~ λ2d


BNC Neutron school

Supermirror

26

d1}

λ 1λ 2λ 3λ 4

λ c

d2} d3} d4}


BNC Neutron school

He3

27

He3 absorbs only neutrons with spin antiparallel to the nuclear spin


BNC Neutron school

Instruments

28

Instruments


BNC Neutron school

D7

29


BNC Neutron school

Cryopad

30


BNC Neutron school

NSE basics

312

Selector

Polarizer

pi/2 flipper

Precession(1)

pi flipper

Precession(2)

pi/2 flipper

Analyser

Detector

Beam polarisation

Magnetic field direction

Sample

ωL = γL ·B

�

B1dℓ =π / 2

π

∫ B2dℓπ

π / 2

∫

Echo condition: The measured quantity is: S(q,t)/S(q,0) where

�

t ∝λ3 Bdℓ∫

= 0ϕtot = γB1l1v1 �

γB2l2v2For elastic scattering:

ϕtot = hγBlmv3ω+o

✓⇣ω

1/2mv2⌘2◆For omega energy

exchange:

S(q,ω)The probability of omega energy exchange:

hcosϕi =Rcos(hγBl

mv3ω)S(q,ω)dω

RS(q,ω)dω = S(q, t)The final polarization:


BNC Neutron school

resolution correction

32

1.0

0.8

0.6

0.4

0.2

0.0

Echo

2520151050time [nsec]

sample (resolution corrected)

1.0

0.8

0.6

0.4

0.2

0.0

Echo

2520151050time [nsec]

sample resolution

As mesures the fourier transforme

S(q,t) the instrumental resolution is a

simple division instead of deconvolution


BNC Neutron school

When NOT to use NSE

33

Say you have a weak but well defined excitation

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00

Intensity

-100 -50 0 50 100omega

S (q,w) = 0.95 · d(w)+0.025 · (d(w�w0)+d(w+w0))

1.0

0.8

0.6

0.4

0.2

0.0

echo

100806040200time

S (q, t) = 0.95+0.05 · cos(w0

)

Bad signal to noise, Δλ/λ => no good for Spin Echo


BNC Neutron school

Reptation

34

1.0

0.8

0.6

0.4

0.2

0.0

S(q,t)

0.120.100.080.060.040.020.00

q2sqrt(t)

q values

0.49 nm-1

0.78 nm-1

0.97 nm-1

1.16 nm-1

1.36 nm-1

1.0

0.8

0.6

0.4

0.2

0.0

S(q,t)

3530252015105

time [nsec]

0.49 nm-1

0.78 nm-1

0.97 nm-1

1.16 nm-1

1.36 nm-1

10% marked polymer chain(H) in deuterated matrix of the same polymer melt

at short time => Rouse dynamics 1/tau ~ q4 at longer times starts to feel the “tube” formed by the other chains (deGennes)

1.0

0.8

0.6

0.4

0.2

0.0

S(q,t)

160140120100806040200time [nsec]

1.0

0.8

0.6

0.4

0.2

0.0

S(q,t)

0.250.200.150.100.05

q^2sqrt(time)

Q values

0.50 nm-1

0.77 nm-1

0.96 nm-1

1.15 nm-1

1.45 nm-1

D. Richter, B. Ewen, B. Farago, et al., Physical Review Letters 62, 2140 (1989).

P. Schleger, B. Farago, C. Lartigue, et al., Physical Review Letters 81, 124 (1998).


BNC Neutron school 35

Spin Ice

‘spin ice’ materials Ho2Ti2O2 , Dy2Ti2O7 and Ho2Sn2O7 the spin is equivalent to the H displacement vector in water ice

Letter to the Editor L11

Figure 1. NSE results for Ho2Ti2O7: (a) integrated over all Q and fitted to an exponential function;(b) as a function of Q, indicating negligible Q-dependence.

Grenoble, using a neutron wavelength of λ = 5.5 Å. The spectrometer was calibrated witha reference sample that does not show dynamics on the timescale of this experiment. At alltemperatures between 0.05 and 200 K the measured relaxation can be fitted with excellentprecision to a simple exponential function:

F(Q, t) = (1 − B) exp{−"(T )t}, (1)

where B = 0.09 ± 0.01. The fact that B is finite, that is F(Q, t) = 1 at ∼10−12 s, provesthe existence of relaxation processes at short timescales beyond the resolution of the NSEtechnique. We suggest that these very fast processes might be associated with small incoherentoscillations of the spins about their ⟨111⟩ easy axes. Such processes cannot relax the moreimportant magnetic fluctuations associated with spin reversals, and hence are unable to providecomplete relaxation of the system. We do not comment on them further in this letter.

The frequency "(T ) can be fitted to an Arrhenius expression "(T ) = 2"h exp(−Ea/kT )

with attempt frequency "h = 1.1 ± 0.2 × 1011 Hz (∼5 K) and activation energy Eh =293 ± 12 K. The relaxation was found to be Q-independent (figure 1(b)), an important clue toits origin (see below). For the rest of the letter we shall refer to this exponential relaxation asthe ‘high’ process, referring to its typical temperature range, with subscript ‘h’.

Single exponential thermally activated

Letter to the Editor L11

Figure 1. NSE results for Ho2Ti2O7: (a) integrated over all Q and fitted to an exponential function;(b) as a function of Q, indicating negligible Q-dependence.

Grenoble, using a neutron wavelength of λ = 5.5 Å. The spectrometer was calibrated witha reference sample that does not show dynamics on the timescale of this experiment. At alltemperatures between 0.05 and 200 K the measured relaxation can be fitted with excellentprecision to a simple exponential function:

F(Q, t) = (1 − B) exp{−"(T )t}, (1)

where B = 0.09 ± 0.01. The fact that B is finite, that is F(Q, t) = 1 at ∼10−12 s, provesthe existence of relaxation processes at short timescales beyond the resolution of the NSEtechnique. We suggest that these very fast processes might be associated with small incoherentoscillations of the spins about their ⟨111⟩ easy axes. Such processes cannot relax the moreimportant magnetic fluctuations associated with spin reversals, and hence are unable to providecomplete relaxation of the system. We do not comment on them further in this letter.

The frequency "(T ) can be fitted to an Arrhenius expression "(T ) = 2"h exp(−Ea/kT )

with attempt frequency "h = 1.1 ± 0.2 × 1011 Hz (∼5 K) and activation energy Eh =293 ± 12 K. The relaxation was found to be Q-independent (figure 1(b)), an important clue toits origin (see below). For the rest of the letter we shall refer to this exponential relaxation asthe ‘high’ process, referring to its typical temperature range, with subscript ‘h’.

Q independent relaxation

G. Ehlers , Cornelius ,A L, Orendac ,M, Kajnakova ,M, Fennell ,T, Bramwell ,S T, Gardner , Journal of Physics Condensed Matter 15, L9 (2003).

Paramagnetic echo => Only magnetic scattering gives echo !


BNC Neutron school

Over

36

It’s over folks ! thanks...

will be presented Outline - Oxford NeutronSchool · B. Farago Polarized Neutrons BNC Neutron school Isotop and spin incoherence 4 bi can have a distribution because: - different isotopes

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