Joint Research Centre - unimi.itscuolaaimagn2016.fisica.unimi.it/lessons/Caciuffo.pdf · with...

2016 AIMag School, Milano 18-22 April 2016 roberto.caciuffo@ec.europa.eu

Joint Research Centre The European Commission’s in-house science service

Neutrons: a quantum probe for magnetism

Roberto Caciuffo

EC JRC, Institute for Transuranium Elements,

Karlsruhe (Germany)

roberto.caciuffo@ec.europa.eu

2

• Electrically neutral

• Microscopically magnetic

• Sensitive to light atoms

Neutrons – a tailor-made probe

H Li C O S Mn Zr Cs

X-rays

neutrons

mass mN = 1.67510-27 kg

charge qN = 0; spin = 1/2

magnetic dipole moment n = -1.913N

k = 2 = mNv/

E(meV) = 2.072 k2 (Å-1) = 81.797/2 (Å)

(Å) = 3956/v(ms-1)

3

Neutrons interact with nuclei via the short-range strong nuclear force, and with electrons via a dipole-dipole interaction between magnetic moments.

The interaction of neutrons with matter is weak: • Theoretically easy to model • Large penetration depth

Neutrons penetrate matter much more deeply than X-rays or charged particles,

as the size of neutron scattering centres

is typically 10-5 times smaller than the distance between those centres.

Neutrons – a gentle probe

4

100

104

108

1012

1016

1020

1024

1028

1032

1036

Pe

ak B

rilli

ance

(pa

rtic

les s

-1 m

rad

-2 m

m-2

0.1

% B

W)

Neutrons – a scarce and expensive probe

5

Weak interactions combined with low fluxes make neutron scattering a signal-limited technique. Its use is justified by the uniqueness of the information it provides.

a probe that lets you see different things...

making obvious the unexpected...

Neutrons – a unique probe

Courtesy P. G. Radaelli, Oxford University

Neutrons with energy comparable with that of elementary excitations in condensed matter have a wavelength matching interatomic distances.

E(meV) T(K) (Å)

Cold 0.1-10 1-120 0.4-3

Thermal 5-100 60-103 0.1-0.4

Hot 102-103 103-104 0.04-0.1

• Appropriate length and energy scale

New chemicals

Novel materials

Fundamental understanding of Nature

This makes energy-, momentum-, and spin-resolved neutron scattering a powerful tool for probing structure and dynamics of materials at different level of complexity, energy and length scales.

Neutrons – probing structure and dynamics of condensed matter

7

Hard

Soft multidisciplinary condensed matter science

1960

1970

1990

1980

2000

1960

1970

1990

1980

No One Experiment

Courtesy Andrew Taylor, STFC

8

Neutron sources: nuclear fission and spallation

Fission

Spallation

9

Pulsed versus Steady State Sources

Spallation 30 MeV/n

Fission 190 MeV/n

200 kW ISIS

58 MW ILL

Inte

ns

ity

time

Pulsed Sources

•More neutrons at high energy

•Neutrons produced in burst

•Pulsed operation

•Sharp pulses give high resolution

•Resolution function asymmetric

•Must use time of flight methods

•Horizons still to be explored

•Seen as environment friendly

Nuclear Reactors

•More neutrons at low energy

•Easier to shield

•Continuous operation

•Resolution can be adapted

•Resolution function Gaussian

•More flexibility

•Further development limited

•Seen as environmen unfriendly

10

Scattering experiments

Momentum transfer

Energy transfer

Definition of cross-section:

[s] = barn (1 barn = 10-28 m2)

s is the effective area presented by the target particle

to the passing neutron, which is scattered if it hits this area. s is then related to the scattering probability.

2

S

D

∆𝐼 = 𝑁Φ 𝐸𝑖 ∆𝐸𝑖𝑑2𝜎

𝑑Ω𝑑𝜔 ∆ΩΔ𝐸𝑓

𝑸 = (𝒌𝑖 − 𝒌𝑓) 𝒌𝑓 ± ∆𝒌𝑓

𝐸𝑓 ± ∆𝐸𝑓

𝑷𝑓

𝒌𝑖 ± ∆𝒌𝑖

𝐸𝑖 ± ∆𝐸𝑖

𝑷𝑓

∆𝐼 = neutrons recorded per unit time

𝑁 = number of scattering centres

= incident flux (n/time/area)

DW= accepted solid angle

DEf = accepted energy window

𝜔 = 𝐸𝑖 − 𝐸𝑓 = 2

2𝑚 (𝑘𝑖

2 − 𝑘𝑓2)

𝑁Φ 𝐸𝑖 ∆𝐸𝑖

11

Elastic scattering Inelastic scattering

ki kf

-kf

Q

Scattering triangle

Structure of

materials

Dynamics of

materials

𝐸𝑖 = 𝐸𝑓 𝑘𝑖 = 𝑘𝑓 = 2𝜋/𝜆

𝑄 = 2𝑘𝑖 sin𝜃 𝑄2 = 𝑘𝑖2 + 𝑘𝑓

2 − 2𝑘𝑖𝑘𝑓 𝑐𝑜𝑠𝜙

𝐸𝑖 ≠ 𝐸𝑓 𝑘𝑖 ≠ 𝑘𝑓

ki

kf

Q

12

𝑸 = 𝒌𝑖 − 𝒌𝑓

𝑄𝑥 = 𝑘𝑖 − 𝑘𝑓𝑠𝑖𝑛𝜗 𝑐𝑜𝑠𝜑

𝑄𝑦 = − 𝑘𝑖𝑠𝑖𝑛𝜗 𝑠𝑖𝑛𝜑

𝑄𝑧 = − 𝑘𝑓 𝑐𝑜𝑠𝜑

x

y

z

ki

kf

Q

Ei = 300 meV Ei = 600 meV

=2 deg

=110

D=4

Kinematical constraints in measuring scattering processes

Each detector traces a parabolic trajectory trough (Q,w) space

13

Selecting neutrons energy

Crystal monochromators

2dsin = n

∆𝜆

𝜆

2

= 𝛼𝑡𝑜𝑡 𝑐𝑜𝑡𝜃 2 +

∆𝑑

𝑑

2

Velocity selectors and choppers

G=2/d

G/2

14

Counting neutrons

Detectors tank of LET@ISIS

3He linear PSDs of MERLIN @ISIS (3 m long, 69632 pixels)

nuclear reactions used to convert neutrons into charged particles: • gas proportional counters • ionization chambers • scintillation detectors • semiconductor detectors

Scintillator detector Module of GEM @ISIS

prototype of a Gas-Electron Multiplier detector using 10B4C-coated alumina lamellae for LOKI @ESS (G. Gorini)

15

Manipulating the neutron spin

• The neutron wavefunction is a two-component spinor

• The neutron-nucleus interaction VN is spin-dependent

• The magnetic interaction between neutrons and atomic magnetic moment is spin dependent

Polarization can be achieved by spin-dependent absorption from a nuclear spin polarized target (3He spin filter), by scattering from a ferromagnetic crystal or by reflection by a birefringent medium

Ψ = Ψ + Ψ = 𝑐𝑒𝑖𝑘 ∙𝑟

10 + 𝑐𝑒

𝑖𝑘 ∙𝑟 01

𝑘 , = 𝑛, 𝑘 0

𝑛,2 = 1 −

2𝑚

( k0)2 𝑉𝑁 ± 𝜇𝐵 = 1 −

4𝜋

k02 𝜚𝑁 ± 𝜚𝐵

𝑠 =

2𝜎 =

2

0 11 0

, 0 −𝑖𝑖 0

, 1 00 −1

𝜇 = −1.913e

2𝑚 𝜎 = 𝛾𝐿𝑠

𝛾𝐿 = −1.832 × 108 𝑟𝑎𝑑 𝑠−1𝑇−1

16

Manipulating the neutron spin

Beam polarization vector P: statistical average of an ensemble of spins 𝑃 = < 𝜎 > = 𝑇𝑟(𝜚 𝜎 )

𝜚 = 1

2

1 00 1

+ 𝜎 ∙ 𝑃 = 1

2

1 + 𝑃𝑧 𝑃𝑥 − 𝑖𝑃𝑦𝑃𝑥 + 𝑖𝑃𝑦 1 − 𝑃𝑧

Component of P along an arbitrary direction n:

𝑃𝑛 = 𝑇𝑟[𝜚 (𝑛𝑥𝜎𝑥 + 𝑛𝑦𝜎𝑦 + 𝑛𝑧𝜎𝑧)]

Experimentally one measures the component of P along an applied magnetic field:

𝑃 = 𝐼 − 𝐼𝐼 + 𝐼

17

Manipulating the neutron spin

Heussler Cu2MnAl monochromators. Spin-dependent Bragg diffraction (95% polarization)

Polarized 3He filters. Spin-dependent absorption (~99-100% polarization)

Fe/Si supermirrors. Spin-dependent reflection(96-99% polarization)

© ILL

18

Manipulating the neutron spin

In a magnetic field B, the neutron feels a potential

𝜔𝐿 𝑟𝑎𝑑 𝑠−1 = 18325 𝐵(𝐺)

𝑉𝑚 = −𝜇 ∙ 𝐵 = −𝛾𝐿𝟐

𝜎𝑥𝐵𝑥 + 𝜎𝑦𝐵𝑦 + 𝜎𝑧𝐵𝑧 = −𝛾𝐿𝟐

𝐵𝑧 𝐵𝑥 − 𝑖𝐵𝑦

𝐵𝑥 + 𝑖𝐵𝑦 −𝐵𝑧

The polarization undergoes a Larmor precession with an angular frequency

© ILL

19

If the direction of B rotates with a frequency wB << wL , P is

transported adiabatically (𝜇 ∙ 𝐵 conserved)

Manipulating the neutron spin

Otherwise 𝜇 ∙ 𝐵 is not conserved

For a /2 rotation over 5 cm = 1 Å B = 204 G = 10 Å B = 20 G

20

Courtesy of E. Lelièvre-Berna, ILL

Spherical polarimetry with CRYOPAD

21

Instruments without energy analysis (diffractometers)

WISH@ISIS

D9@ILL

D3@ILL

Atomic and magnetic structures

Crystals

Gasses

Liquids

Amorphous solids

VIVALDI@ILL

GEM @ ISIS

D2B @ ILL

22

D22@ILL

SANS2D@ISIS

Small Angle Neutron Scattering

Measuring structures on the scale of 1 to 100 nm

Flux lattices in superconductors

Magnetic correlations in

nanoparticles

23

Neutron reflectometry

• Magnetic thin films and nanostructures (m-nm)

• Spintronic

• Magnetization at the interface

• Spin injection

• Kinetics (s-ms)

• Excitations

CRISP@

ISIS

FIGARO@ILL

Characterization of interfacial phenomena on the microscopic length-

scale

Fe

Si

Vn

Vn + Vm

Vn - Vm

24

Three-axis spectrometers

Collective motion of

atoms and magnetic

moments in single

crystalline samples.

IN20@ILL

𝑄2 = 𝑘𝑖2 + 𝑘𝑓

2 − 2𝑘𝑖𝑘𝑓 𝑐𝑜𝑠𝜙 kf ki

Q

G

q

kf

Constant-Q scan at fixed Ei Constant-E scan

𝑘𝑖 = 𝜋

𝑑𝑀𝑠𝑖𝑛𝜃𝑀

𝑘𝑓 = 𝜋

𝑑𝐴𝑠𝑖𝑛𝜃𝐴

𝜔 ≥2𝑄2

2𝑚 𝑠𝑖𝑛𝜙

25

Time-of-Flight Chopper Spectrometers

MERLIN@ISIS

LET@ISIS

IN5@ILL

26

Time-of-flight measurements

Direct geometry Indirect geometry

Time

source

monochromator

sample

detector

Time

source

E analyzer

sample

detector

source

chopper

detector L0

L1 L2

sample

Fixed initial energy All final energies -∞< w<Ei

source detector

L 0

L 1

L 2

2

analyzer

sample

All incident energies Fixed final energy -Ef < w<∞

ti t

𝐸𝑓 = 1

2 𝑚

𝐿22

(𝑡 − 𝑡𝑖)2

𝑘𝑓 = 𝜋

𝑑𝐴𝑠𝑖𝑛𝜃𝐴

𝑘𝑖 = 𝜋 𝑚 𝐿0

𝜋𝑡 − 𝑚 (𝐿1+𝐿2) 𝑑𝐴𝑠𝑖𝑛𝜃𝐴

t=0

E(meV) = 5.227106 t-2 (m2/s2)

3 m

MERLIN – Large detector Coverage

• steradians of solid angle

• 3m long position sensitive detectors

• position resolution along tube ~ 20mm

• Total of 69000 ‘pixel’ elements

• 2000 time channels/pixel (14x107 bins)

• detectors in evacuated 30 m3 tank

• virtually gap free coverage

±30o

+135o -45o 3o

28

(meV)

Inte

nsity (

arb

. units)

0 2 4 6 8 100

100

200

300

400

500

•Visualisation software

Combine ~200 datasets full map of S(Q,) Bespoke visualisation

40GB 109 pixels software (“HORACE”)

29

Spin-Echo Spectrometers

slow relaxation in magnetic materials elastic paramagnetic scattering phonon linewidths

IN11@ILL

Neutron energy encoded into

a Larmor precession angle

Energy range: 1.3x10-5 ... 0.01 meV

Interaction of a neutron with a nucleus at position Rj:

Neutron-nucleus scattering

The Fermi length b has different values for neutron and nucleus spins parallel or antiparallel

b(238U) = 0.8417 x 10-12 cm b(H) = -0.374 x 10-12 cm

Coherent scattering due to the “average” nucleus; intensity as the square of the sum of individual amplitudes: interference Incoherent scattering due to deviation from the average “nucleus”; intensity as the sum of individual intensities: no interference

𝑉𝑁 = 2𝜋2

𝑚 𝑏𝑗 𝛿(𝑟 − 𝑅 𝑗 )

𝑏𝑗 = 𝐼𝑗 + 1 𝑏𝑗

↑ + 𝐼𝐽𝑏𝑗↓

2𝐼𝑗 + 1+

𝑏𝑗↑ − 𝑏𝑗

↓

2𝐼𝑗 + 1 𝜎 ∙ 𝐼 𝑗

31

Neutron interaction with a magnetic field

𝑉𝑚 = −𝜇 ∙ 𝐵 = −𝛾𝐿𝟐

𝜎𝑥𝐵𝑥 + 𝜎𝑦𝐵𝑦 + 𝜎𝑧𝐵𝑧 = −𝛾𝐿𝟐

𝐵𝑧 𝐵𝑥 − 𝑖𝐵𝑦

𝐵𝑥 + 𝑖𝐵𝑦 −𝐵𝑧

The magnetic field that scatters the neutrons is due to currents and magnetic dipole moments of electrons.

𝐵 𝑒 = 2 𝜇𝐵 (𝑐𝑢𝑟𝑙𝑠 × 𝑟

𝑟3+

1

𝑝 × 𝑟

𝑟3)

The total field B(r,t) in a sample is the sum of the fields generated by all the electrons and depends on the wavefunction of the system.

e

n

R

r

ri

s

p

32

Scattering cross section

Born approximation and Fermi golden rule

𝑑𝜎

𝑑𝛺𝑑𝜔 𝑘𝑖𝜎𝑖→𝑘𝑓𝜎𝑓

= 𝑘𝑓

𝑘𝑖 𝑚

2𝜋ℏ

2

𝑝𝑓𝑝𝑖 Ψ𝑓 𝑉 Ψ𝑖 2

𝑖 ,𝑓

𝛿(ℏ𝜔 + 휀𝑓 − 휀𝑖)

Ψ ∝ exp(𝑖 𝑘 ∙ 𝑟 ) σ Υ𝑡𝑎𝑟𝑔𝑒𝑡

𝑘𝑓 𝑉 𝑘𝑓 = 𝑉( 𝑟 ) exp −𝑖 𝑘 𝑓 − 𝑘 𝑖 ∙ 𝑟 𝑑𝑟 = 𝑉( 𝑟 ) exp −𝑖 𝑄 ∙ 𝑟 𝑑𝑟

Born approximation and Fermi golden rule:

The matrix elements contains the physics of the neutron-sample system with wavefunction

The scattering amplitude is the Fourier transform of the interaction potential in the reciprocal space

33

For the magnetic interaction the scattering amplitude aM(Q) is:

Q

s

sQ

QsQ

Q

p

pQ

Magnetic scattering amplitude

−𝜇0𝑀 ⊥ 𝑄 = 𝐵 𝑟 exp 𝑖 𝑄 ∙ 𝑟 𝑑𝑟

Ψ𝑓 −𝜇 ∙ 𝐵 Ψ𝑖 = 𝜇0 𝑠𝑓 𝜇 𝑠𝑖 Υ𝑓 𝑀 ⊥ 𝑄 Υ𝑖

The total field B(r, t) in a sample of condensed matter is the sum of the fields generated by all the electrons. Defining

𝑎𝑀 𝑄 = 𝑘𝑓 𝑉𝑚 (𝑟 ) 𝑘𝑓 = 4𝜋

𝑄2exp 𝑖 𝑄 ∙ 𝑟 𝑄 × 𝑠 × 𝑄 +

4𝜋𝑖

𝑄2exp 𝑖 𝑄 ∙ 𝑟 𝑝 × 𝑄

𝑑𝜎

𝑑𝛺𝑑𝜔 𝑘𝑖↑→𝑘𝑓𝜎𝑓

= 𝑘𝑓

𝑘𝑖 𝛾𝑟0

2𝜇𝐵

2

4𝜋 2 𝑝𝑓𝑝𝑖 λ𝑓 𝑀 ⊥ 𝑄 λ𝑖 ⋅ 𝑠𝑓 𝜎 ↑ 2

𝑖 ,𝑓

𝛿(휀𝑓 − 휀𝑖 − ℏ𝜔)

34

magnetic scattering amplitude

The magnetic scattering of neutrons depends only on the component

of the magnetisation perpendicular to the scattering vector Q.

Q

M(q)

M

MQ

The magnetic neutron scattering cross section measures correlations in

magnetization, that is how the magnetization on a given site influence the

magnetization of the surrounding.

𝑑𝜎

𝑑𝛺𝑑𝜔 ∝ 𝑀 ⊥

∗ 𝑄 𝑀 ⊥ 𝑄

35

𝑑𝜎

𝑑𝛺𝑑𝜔 ∝ 𝑀 ⊥

∗ 𝑄 𝑀 ⊥ 𝑄

Flux Line Lattice in superconductors

SANS

ki

kf

Q = Ki-Kf almost to M J. S. White et al., PRL 102, 097001 (2009)

B

Diffraction patterns from the FLL in YBa2Cu3O7, as a function of field applied perpendicular to the CuO2 planes. SC in the chains is suppressed by increasing field and the d-wave nodes move closer to 45° angles.

𝜇 = −𝜇𝐵 𝐿 + 2𝑆 = −𝑔𝐽𝜇𝐵 𝐽

𝑎𝑀 𝑄 = −𝑝𝑔𝐽𝑓 𝑄 𝐽 ⊥ ∙ 𝜎

Russell-Sanders coupling, ground multiplet J:

f(Q) is the magnetic form factor. It arises from the spatial distribution

of unpaired electrons around a magnetic atom. If r is the normalized

density of spin around the equilibrium position:

𝑓 𝑄 = 𝜌↑ 𝑟 − 𝜌↓ 𝑟 exp 𝑖 𝑄 ∙ 𝑟 𝑑𝑟 𝑎𝑡𝑜𝑚 𝑣𝑜𝑙𝑢𝑚𝑒

magnetic form factor

r

Pu3+ Pu3+

Fe3+

𝑓 𝑄 = 𝑗0(𝑄) +𝐿

𝐿 + 𝑠 𝑗2(𝑄)

Pu has 5 5f electrons:

S=–5/2; L=5; J=5/2

Strong cancellation of spin and

orbital components.

Almost equal contributions

from <jo> and <j2>

37

Elastic scattering from a crystal.

Constructive interference results in Bragg peaks

only if 𝑄 is equal to a reciprocal lattice vector 𝜏

𝑑𝜎

𝑑𝛺=

𝑑𝜎

𝑑𝛺𝑑𝜔 𝑑𝜔 = 𝑁

2𝜋 3

𝑉0 𝐹𝑁 𝑄

2

𝜏

𝛿(𝑄 − 𝜏 )

𝜏 = ℎ𝑎 ∗ + 𝑘𝑏 ∗ + ℓ𝑐 ∗

FN are the coefficients of the Fourier transform of the particle density.

A Fourier synthesis is possible if FN are known in amplitude and phase. However, measurements give

𝐹𝑁 𝑄2 not amplitude and phase.

𝐹𝑁 𝑄 = 𝑏𝑑𝑑

𝑒−𝐵𝑄2 𝑒2𝜋𝑖 (ℎ𝑥𝑑+𝑘𝑦𝑑+ℓ𝑧𝑑 )

ki kf

Q

38

The phase problem

The dog b

and its F-transform

A duck

and its F-transform

-F[b] m-F[b]

-F[duck] m-F[duck]

anti F-transform

anti F-transform

-F[b] m-F[b]

-F[duck] m-F[duck]

39

The phase problem

-F[b]

m-F[b] -F[duck]

m-F[duck]

anti F-transform

anti F-transform

As phases are not measured, structural determinations require parametric modelling

40

Magnetic crystallography

Magnetic structure factor (vector)

𝐹 𝑀 𝑄 = 𝑝 𝑚 𝑑𝑑

𝑓 𝑄 𝑒−𝐵𝑄2 𝑒2𝜋𝑖 (ℎ𝑥𝑑+𝑘𝑦𝑑+ℓ𝑧𝑑 )

𝑑𝜎𝑀𝑑𝛺

= 𝑁 2𝜋 3

𝑉0 𝐹𝑀⊥ 𝑄

2

𝜏

𝛿(𝑄 − 𝜏 𝑀)

𝑚 𝑣𝑑 = 𝜇 𝑣𝑑 = 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑐 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑎𝑡𝑜𝑚 𝑑 𝑖𝑛 𝑐𝑒𝑙𝑙 𝑣

The magnetic moment distribution can be Fourier expanded

𝑚 𝑣𝑑 = 𝑚 𝑘𝑑 exp −𝑖 𝑘 ∙ 𝑅 𝑣

𝑘

k is the propagation vector of the magnetic structure. Only k

vectors confined within the first Brillouin zone of the Bravais

lattice of the crystallographic cell enter into the summation.

41

The propagation vector describes the propagation direction and

Wavelength of the ordering waves: it defines the relation between

the moments in neighbouring cell of the crystal.

The set of symmetry equivalent k vectors is defined as {k} and is

called the vector star. Each individual k in the set is called an arm of

the star.

The propagation vector

𝑚 𝑣𝑑 = 𝑚 𝑘𝑑 exp −𝑖 𝑘 ∙ 𝑅 𝑣

𝑘

𝑑𝜎𝑀𝑑𝛺

= 𝑁 2𝜋 3

𝑉0 𝐹𝑀⊥ 𝑄

2

𝜏

𝛿(𝑄 − 𝑘 − 𝜏 𝑀)

𝑘

Magnetic Bragg peaks occur at

𝑄 = 𝑘 + 𝜏 𝑀

Within the Brillouin zone defined by there is a number of magnetic peaks equal to the number of distinct wave vectors k in the sum

42

Fluorite-type structure Long-range order (Jones et al. 1952, Osborne & Westrum 1953)

0

4

8

12

300

340

380

0 40 80 120 160

Cp (

cal K

-1 m

ol-1

)

Temperature (K)

Magnetic Structure of UO2

Propagation vector k = (0, 0, 1) {k} = (1, 0, 0) (0, 1, 0) (0, 0, 1)

43

Type I, 3-k transverse structure 0 = 1.74 B

n m exp(i kj•Rn) kj

j = 1

j = 3

k1 = (1, 0, 0) etc.

Magnetic Structure of UO2

(0 0 0) m1 = (1 1 1) (1/2 1/2 0) m2 = (1 -1 -1) (1/2 0 1/2) m3 = (-1 -1 1) (0 1/2 1/2 ) m4 = (-1 1 -1)

(0 0 0) m1 = (1 1 1) (1/2 1/2 0) m2 = (-1 1 -1) (1/2 0 1/2) m3 = (1 -1 -1) (0 1/2 1/2 ) m4 = (-1 -1 1)

𝐷𝑜𝑚𝑎𝑖𝑛 𝐵: 𝑚 100 = 0, 0, 1 𝑒𝑡𝑐. 𝐷𝑜𝑚𝑎𝑖𝑛 𝐴: 𝑚 100 = 0, 1, 0 𝑒𝑡𝑐.

44

A cold-neutron high-resolution powder neutron diffractometer:

WISH@ISIS

d-spacing range: 0.7-50 Å

LC Chapon et al, Neutron News 22, 22 (2011)

a) the magnetic structure can have a large number of degrees of freedom (three components of the magnetic moment on several inequivalent atoms);

b) the d-range available to observe Bragg peaks is limited due to fall-off of f(Q) c) magnetic and nuclear Bragg peaks are often nearly overlapping d) powder averaging of the magnetic structure factor for quasi-degenerate reflections

may prevent the determination of the direction of the magnetic moments:

high-resolution data are required to solve the structure.

Long range magnetic order of the quasicrystal approximant in the Tb-Au-Si system (G. Gebresenbut et al., JPCM, 26, 322202 (2014))

45

Nature 442, 797-801 (2006)

Complex magnetic structures

46

Polarised neutron diffraction

Unpolarised neutrons

𝐼 𝑄 = 𝐹𝑁(𝑄 ) 2

+ 𝐹 𝑀⊥ 𝑄 2

𝐼 𝑄 = 𝐹𝑁(𝑄 ) 2

+ 2𝑃 ∙ 𝐹 𝑀⊥ 𝑄 𝐹𝑁 𝑄 + 𝐹 𝑀⊥ 𝑄 2

Polarised neutrons

𝐼+ = 𝐹𝑁 𝑄 + 𝐹𝑀(𝑄 ) 2

𝐼− = 𝐹𝑁 𝑄 − 𝐹𝑀(𝑄 ) 2

𝑃 = 𝑛↑ − 𝑛↓ 𝑧

𝑅 = 𝐼+

𝐼−=

1 + 𝛾

1 − 𝛾

2

𝛾 = 𝐹𝑀𝐹𝑁

𝐹 𝑀 = 𝐹𝑀 𝑧

𝑃 = ±1, 𝑄 ⊥ 𝑃

47

F D

S

B I+

I-

Polarised neutron diffraction

For small ,

Ex.: = 0.1

With unpolarised neutrons

𝑅 ≃ 1 + 4𝛾

𝑅 = 𝐼+

𝐼−=

1 + 𝛾

1 − 𝛾

2

𝐼+ = 𝐹𝑁 𝑄 2

1 + 𝛾 2 = 1.21 𝐹𝑁 𝑄 2

𝐼− = 𝐹𝑁 𝑄 2

1 − 𝛾 2 = 0.81 𝐹𝑁 𝑄 2

𝐼 = 𝐹𝑁 𝑄 2

1 + 𝛾2 = 1.01 𝐹𝑁 𝑄 2

48

Magnetic form factor of NpCoGa5

dipole approximation

f(Q) = [<j0> + C2 <j2>]

C2 = (2-g)/g = L/(L+ S)

C2 = 2.11 Np3+ I.C.

= 0.091(1) B (B = 9.6 T) (Å-1)

magnetic f

orm

facto

r (

B/f

.u.

𝑗𝑛(𝑄) = 𝑈2(𝑟 )∞

0

𝑗𝑛 𝑄𝑟 𝑑𝑟

49

Structure of the Cr8Cd molecular ring. Cr atoms are represented in green, Cd in purple, O in red, F in yellow and C in black. H ions are omitted for simplicity.

Finite size effects in chains of antiferromagnetically coupled spins

Top: Spin density map for Cr8Cd (scale in μB Å

−2) obtained by the refinement of PND experimental data for applied fields 9 tesla at T=1.8 K (projection along the crystal b axis). Bottom: Classical spin ground state configuration for an even-open chain (a) and an even-closed chain (b) of AF-coupled spins under an external magnetic field.

T. Guidi et al., Nat. Commun. 6:7061 doi: 10.1038/ncomms8061 (2015).

50

Magnetic inelastic neutron scattering

• Single ion excitations

• crystal field measurements

• quasi 0-D clusters of spins

• 1-D spin chains, 2-D square lattices, 3-D systems

• frustrated magnetic systems……

INS offers the ability to measure directly the interactions of magnetic moments with other magnetic moments and with the local environment. A variety of problems can be investigated in a variety of systems:

Magnetic inelastic neutron scattering (INS) probes spin dynamics

The partial differential cross section at non-zero temperature is

proportional to the space-time Fourier transform of a spin-spin

correlation function

51

Magnetic inelastic neutron scattering cross-section

The cross-section for scattering of n0 in the |> initial spin state into the final spin state <sf| at T = 0 is (r0 = 0.5410-12 cm):

𝑑𝜎

𝑑𝛺𝑑𝜔 𝑘𝑖↑→𝑘𝑓↑

= 𝑘𝑓

𝑘𝑖 𝛾𝑟0

2𝜇𝐵

2

4𝜋 2 𝑝𝑓𝑝𝑖 λ𝑓 𝑀 ⊥𝑧 λ𝑖 2

𝑖 ,𝑓

𝛿(휀𝑓 − 휀𝑖 − ℏ𝜔)

𝑑𝜎

𝑑𝛺𝑑𝜔 𝑘𝑖↑→𝑘𝑓↓

= 𝑘𝑓

𝑘𝑖 𝛾𝑟0

2𝜇𝐵

2

4𝜋 2 𝑝𝑓𝑝𝑖 λ𝑓 𝑀 ⊥𝑥 λ𝑖 + 𝑖 λ𝑓 𝑀 ⊥𝑦 λ𝑖 2

𝑖 ,𝑓

𝛿(휀𝑓 − 휀𝑖 − ℏ𝜔)

Non Spin Flip scattering probes the components of M along the quantization axis of the n0 spin, while Spin Flip scattering probes the components of M perpendicular to it.

Using the Fourier representation of the function, the cross section can be written as the F-Transform of a spin-spin time correlation function

𝑆 ⊥ ∙ 𝑆 ⊥ 𝑄 ,𝜔 = 𝑒𝑖𝑄 ∙(𝑟 𝑛 ′−𝑟 𝑛 )

𝑛 ,𝑛′

𝑒𝑖𝜔𝑡∞

0

𝑑𝑡 𝑝(𝐸𝜆) 𝜆 𝑠 ⊥𝑛(0) ∙ 𝑠 ⊥𝑛′(𝑡) 𝜆

𝜆

NSF:

SF:

52

Spin-spin correlation functions

𝑑𝜎

𝑑𝛺𝑑𝜔 ↑

= 𝐴 𝑄 𝑆 ⊥ ∙ 𝑧 𝑆 ⊥ ∙ 𝑧 𝑄 ,𝜔

𝑑𝜎

𝑑𝛺𝑑𝜔 ↓

= 𝐴 𝑄 (𝑆 ⊥)⊥ ∙ (𝑆 ⊥)⊥ 𝑄 ,𝜔 + 𝑖 𝑆 ⊥ × 𝑆 ⊥ ∙ 𝑧 𝑄 ,𝜔

For neutrons with initial spin

(𝑆 ⊥)⊥ = 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑝𝑖𝑛 ⊥ 𝑡𝑜 𝑄 𝑎𝑛𝑑 𝑡𝑜 𝑧

𝐴 𝑄 = 𝑘𝑓

𝑘𝑖 𝛾𝑟0

2𝜋ℏ

2

𝑓(𝑄 ) 2

𝑒−2𝑊(𝑄)

where,

NSF correlations of the S component to the initial direction of P

SF correlations of the S component to the initial direction of P

53

<Sn(0)Sn’

b(t)> is the thermal average of the time dependent spin operator and corresponds to the van Hove correlation function: the probability of finding a spin Sn’ at site n’ and at time t when the spin at position n is Sn at t=0

: quasielastic broadening : intrinsic linewidth

: lifetime

: correlation length

Spin-spin correlation function

Spin-Spin Corr. Function Magnetic neutron cross section

static moment oscillating moment

relaxing moment

Long range order

Short range order

54

Correlation function unpolarised neutrons

or

or, in terms of matrix elements:

Ex. #4. Derive the above result.

𝑑𝜎

𝑑𝛺𝑑𝜔 = 𝐴 𝑄 𝑆 ⊥ ∙ 𝑆 ⊥ 𝑄 ,𝜔

𝑑𝜎

𝑑𝛺𝑑𝜔 = 𝐴 𝑄 𝛿𝛼 ,𝛽 −

𝑄𝛼𝑄𝛽

𝑄2

𝛼 ,𝛽

𝑆𝛼 ,𝛽 𝑄 ,𝜔 , 𝛼,𝛽 = 𝑥,𝑦, 𝑧

𝑆𝛼 ,𝛽 𝑄 ,𝜔 = 𝑒𝑖𝑄 ∙(𝑟 𝑛 ′−𝑟 𝑛 )

𝑛 ,𝑛′

𝑒𝑖𝜔𝑡∞

0

𝑑𝑡 𝑝(𝐸𝜆) 𝜆 𝑠 ⊥𝑛(0) ∙ 𝑠 ⊥𝑛′(𝑡) 𝜆

𝜆

𝑆𝛼 ,𝛽 𝑄 ,𝜔 = 𝑒𝑖𝑄 ∙(𝑟 𝑛 ′−𝑟 𝑛 )

𝑛 ,𝑛′

𝑝(𝐸𝜆) 𝜆 𝑠𝑛𝛼 𝜆′ ∙ 𝜆′ 𝑠𝑛

𝛽 𝜆

𝜆 ,𝜆′

55

Direct measurement of spin-spin correlation functions by four- dimensional inelastic neutron scattering

Cr8 homometallic ring (Cr3+, s = 3/2, S = 0)

IN6 @ ILL

Data from a powder sample

56

Direct measurement of spin-spin correlation functions by four- dimensional inelastic neutron scattering

Data from a single crystal sample

147° h; ±20˚ v

Z

Y

X’

Z’

Horace scan 4D S(Q,w) Step of 1° over 360°

= 5 Å T= 1.5K each 10’

Courtesy T. Guidi STFC-ISIS

57

Direct measurement of spin-spin correlation functions by four- dimensional inelastic neutron scattering

Courtesy T. Guidi STFC-ISIS

p=1

p=2

p=3

M. Baker et al Nature Physics (2012)

58

Squared form factor DW factor

Spin correlation function

Inelastic Magnetic Scattering

For ions with unquenched orbital moment and for q0

geometrical factor

𝑑𝜎

𝑑𝛺𝑑𝜔 =

𝑘𝑓

𝑘𝑖 𝛾𝑟0

2𝜋ℏ

2

𝑓(𝑄 ) 2

𝑒−2𝑊(𝑄) 𝛿𝛼 ,𝛽 −𝑄𝛼𝑄𝛽

𝑄2

𝛼 ,𝛽

𝑆𝛼 ,𝛽 𝑄 ,𝜔

𝑠𝑛𝛼 =

1

2𝑔 𝐽𝑛

𝛼 𝑔 = 1 +𝐽 𝐽 + 1 − 𝐿 𝐿 + 1 + 𝑆(𝑆 + 1)

2𝐽(𝐽 + 1)

𝐽𝑛𝛼 𝑏𝑒𝑖𝑛𝑔 𝑎𝑛 𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 𝑜𝑝𝑒𝑟𝑎𝑡𝑜𝑟

59

For a wide class of systems Sb satisfies useful sum-rules

Detailed balance

Total moment

First moment sum-rule

Inelastic Magnetic Scattering

𝑆𝛼𝛽 𝑄 ,𝜔 = exp ℏ𝜔

𝑘𝐵𝑇 𝑆𝛼𝛽 −𝑄 ,−𝜔

ℏ

𝑑𝑄 𝑆𝛼𝛼

𝛼

𝑄 ,𝜔 𝑑𝑄 𝑑𝜔 = 𝑆(𝑆 + 1)

ℏ2 𝑆 𝑄 ,𝜔 𝜔 𝑑𝜔 = −1

3𝑁 𝐽𝑛𝑛 ′ 𝑠 𝑛 ∙ 𝑠 𝑛 ′

𝑛 ,𝑛 ′

1 − cos𝑄 ∙ (𝑟 𝑛′ − 𝑟 𝑛)

60

The scattering function Sb(q, w) is related to the generalized susceptibility b by the fluctuation-dissipation theorem:

b determines the response of the system to the magnetic field established by the neutron:

We convert inelastic scattering data to b to • Compare with bulk susceptibility data • Analyze the temperature dependence of the response • Compare with theories

Note that:

Generalised susceptibility

𝑆𝛼𝛽 𝑄 ,𝜔 = 𝑁ℏ

𝜋

1

1 − exp −ℏ𝜔𝑘𝐵𝑇

𝐼𝑚𝜒𝛼𝛽 𝑄 ,𝜔

𝑀𝛼 𝑄 ,𝜔 = 𝜒𝛼𝛽 𝑄 ,𝜔 𝐻𝛽 𝑄 ,𝜔

𝜒 𝑄 , 0 = 1

2𝜋𝑖 𝑑𝜔

𝐼𝑚𝜒𝛼𝛽 𝑄 ,𝜔

𝜔

61

Valence-fluctuating ground state of delta-plutonium

Experiments find no static magnetism in -Pu.

LDA + Exact Diagonalization of an impurity Anderson model suggests that Pu has an intermediate-valence state <n5f> = 5.21. Similar results from DMFT.

The hybridized ground state of the impurity is a nonmagnetic singlet (S = L = J = 0) The 5f shell magnetic moment fluctuates in time because of the intermediate valence, but is dynamically compensated by the moment of the conduction electron bath

62

Observation of magnetic fluctuations centred around 84 meV in agreement with theory.

Valence-fluctuating ground state of delta-plutonium

Experiment Theory

Crystal Field excitations in AnO2

E

Isolated magnetic ion with total angular momentum J: full rotational symmetry. the ground state is (2J+1)-fold degenerate.

Magnetic ion embedded in a solid:

local charge symmetry lifts partially or totally the degeneracy of the ground state multiplet.

Crystal Field excitations in UO2

5f2 3H4 Cubic symmetry: 2 parameters determine the CF

J = 4

3x

2x

3x

1x

PRB 40, 1865 (1989)

65

Crystal Field excitations in NpO2

5f3 4I9/2

J = 9/2

8

8

62x

4x

4x

E (meV)

5-ph

3x

270

54

0

CF-phonons bound state

0

S (

arb

. units)

Energy Transfer (meV)

2

1

20 40 60 80

Inelastic neutron scattering spectrum at 5K

66

Neptunium ions ground state in NpO2

The Np4+ ground state in the paramagnetic phase of NpO2 is a

quartet of 8 symmetry. In addition to 3 magnetic dipoles and

5 electric quadrupoles, the 8 quartet supports magnetic

octupoles (2, 4, 5), a triplet of magnetic triakontadipoles (5)

and two triplets of rank-7 multipoles degrees of freedom.

67

e-Q primary order parameter m-T primary order parameter

Splitting of the NpO2 ground state

68

Splitting of the NpO2 ground state

INS with polarization analysis on polycrystalline NpO2

Spin Flip (magnetic)

Non Spin Flip (vibrational)

Magnetic Scattering Theory

Splitting of the 15 meV peak due to magnetoelastic interactions?

69

0

5

10

15

20

25

30

0.00 0.25 0.50 0.75 1.00

(,,) (r.l.u.)

En

erg

y (

meV

)

<001> X

LA(D1)

TA(D5)

0

1

2

3

4

5

5 10 15 20

(6,1,0)

(6,0.8,0)

(6,0.6,0)

(6,0.4,0)

(6,0.2,0)

Photo

n In

ten

sity (

arb

. u

nits)

Energy transfer (meV)

0

2

4

6

8

0.4 0.6 0.8 1.0

Inte

nsity (

arb

. u

nits)

reduced wavevector q (rlu)

Magnetoeleastic ph-M25 interactions?

Anomalous behaviour of TA(D5) phonon intensity observed by IXS at energies close to the one predicted for the MM reversal excitation.

Magnetic excit. calculated

IXS measured TA phonon groups

Primary OP: 25 magnetic

multipoles

Magnetic field distribution around a Np ion

Triakontadipole order in NpO2

Lattice dynamics in UO2

Pure spin and quadrupole waves, together with mixed magneto quadrupolar and magneto-vibrational modes. The measured INS cross-section is reproduced by reasonable values of the 5 free parameters.

IN14@ILL

Q = G + q = (1, 1, -1) + (0, 0, )

Transverse constant-Q scans

Lattice dynamics in UO2

Magnon-phonon avoided crossing at q = (0, 0, 0.45)

Longitudinal scan reveals a transverse

50% phonon- 50% magnon mixed mode

Iph (Q·e)2 = 0

eTA

IN22 + CRYOPAD

Quadrupolar modes are visible in the

INS spectra through the associated spin

or vibrational fluctuations.

SA SO QO

SO-TA

TA

Lattice dynamics in UO2

75

INS accounts for the detailed atomic motions and magnetic excitations - individual or collective - within a many-body system.

Microscopic motions or excitations may occur in vastly different time and length scales, typically ps to ms and sub-nm to m: INS necessitates a wide coverage in the energy (E) and wavevector (Q) space with good resolutions.

Interpretation of INS data can be a challenge facing experimentalists. Researchers nowadays have to apply methods of theoretical modeling and simulations that require high degree of sophistication and substantial amount of computing resources.

Concluding remarks