Diffusion des neutrons

Ecole du GDR MICO, Mai 2014 S. Petit (CEA-Saclay)

A quoi servent les neutrons ? Interaction neutron-matière Diffraction Neutrons polarisés Diffusion inélastique

Plan du cours

A quoi servent les neutrons ?

o No charge

o Massive particle

o Spin ½

Do they interact with matter ?

Yes but weakly …

b= scattering length • positive or negative • depends on the isotope

Scattering strength

Interaction forte : le neutron entre au cœur du noyau (« noyau composé »)

Interaction magnétique dipolaire entre le spin du neutron et les moments magnétiques au sein de l’échantillon

Diffraction ?

supposes l ~ a Wavelength of the neutron = a few Å

q

d

Phase difference Bragg law

Spectroscopy ?

supposes E ~ energy of elementary excitations in solids

Energy of thermal neutrons

E = a few meV

a

Detector

Incident neutron flux

Nombre de neutrons dans le détecteur

Thermal neutrons Allow

Diffraction & Spectroscopy

o Structure cristalline, magnétique o Dynamique : relaxation, excitations

collectives

In Europe: Reactors ILL-Grenoble (France) LLB-Saclay (France) FRMII-Munich (Germany) HMI-Berlin (Germany) Spallation sources ISIS-Didcot (UK) PSI-Villigen (Switzerland)

But also: Dubna (Russia), JPARC (Japan) SNS, DOE labs (USA), ANSTO (Australia) Canada, India, …

235U

1n

The reactor provides a spectrum of neutrons with different Ei We need to select an incident energy

9 beam tubes

• 4 thermal

• 2 hot

• 3 cold

9 guides for cold neutrons

Hot source

Cold source

Hot Neutrons Liquid and single crystal structures

Thermal Neutrons Condensed matter

Cold neutrons Condensed matter

q

d

Incident beam

Monochromator

Monochromator

o Should not absorb

o High reflectivity

o d-spacing must be ok

o Large single crystals

o « Cheap » …

Monochromator

Monochromator

Monochromator

Monochromator

How to select an incident energy ?

Other possibilities : Velocity selector

How to select an incident energy ?

Produced a bunch of neutrons with a given incident energy

a = W t = W d/v

v

d

Other possibilities : Disk Chopper

Basic neutron spectrometer

If this is giving you a really hard time, give it a good kick (and you will feel better)

Basic neutron spectrometer

wavevector spin energy Neutrons are plane waves

Incident neutrons

Basic neutron spectrometer

Scattered neutrons

Incident neutrons

Basic neutron spectrometer

Scattering wavevector

Energy conservation

Energy transfer

Scattered neutrons

Incident neutrons

Diffractometer

Incident neutron flux

Scattering angle

1. gain energy (up to infinity) 1. loose energy (up to )

Scattered neutrons in a given solid angle can :

Diffractometer

Most of the neutrons keep their energy : « elastic scattering »

This is the Bragg law !!

Detector

Scattering angle = 2q

Incident neutron flux

Diffractometer

2q

Detector bank

Incident neutron flux

Diffractometer

Unit cell and space group

Inelastic scattering

The neutrons gain or loose energy : « inelastic scattering »

To select a single , one has to « analyze » the scattered beam with an appropiate energy filter Once and are chosen, select by varying a

a

Detector

Incident neutron flux

Inelastic scattering

detector

Incident neutrons

Case 1 : single crystal analyzer

a

Case 1 : single crystal analyzer (Bragg law)

Inelastic scattering

Inelastic scattering

Detector bank

Fast neutrons

Slow Neutrons Elastic response

Incident neutron pulse

Case 2 : « time of flight » analyzer

a

Inelastic scattering

Case 2 : « time of flight » analyzer

a

Incident neutron pulse

Choose Q

Once and are chosen, select by varying a

a

Detector

Incident neutron flux

a

There is just one known wavevector in the lab geometry :

Inelastic scattering

How to select the wavevector for a given energy transfer ?

a

Inelastic scattering

a

Q

Rotate the sample to let coincide Q with Q Useless if the sample is a powder, mandatory if the sample is a single crystal

How to select the wavevector for a given energy transfer ?

Detailed balance

At T=0 K: the system is in its ground state

Ground state

excited state

Absence of symmetry between creation and annihilation processes

Triple axis

Time of flight

Electronics

Control Software

Detector

Positionning

Data storage

Data treatment

An instrument consists in …

Et c’est sans danger …. ou presque

Cross section

Interaction neutron-matière

Cross section

Probability for the incident neutron to be in the si spin state

Probability for the target to be in the initial state l

Density of accessible states kf

Incident flux

Cross section, nuclear interaction

Nuclear interaction with nuclei

b= scattering length • positive or negative • depends on the isotope

Cross section, nuclear interaction

Frozen lattice

Unit cell Atom in the cell

Nothing moves

« simple » definition of the structure factor

Cross section, nuclear interaction

Vibrating lattice

Harmonic approximation

Debye-Waller factor

Cross section, nuclear interaction

Elastic term

Series expansion

Inelastic term : phonons

Cross section, nuclear interaction

Intensity ≠ 0 (Bkg) if ki-kf = Q Ei-Ef = w(Q)

Phonons in PbTe

Cross section, magnetic interaction

Dipolar field created by the spin (and orbital motion) of unpaired electrons

Cross section, magnetic interaction

Unit cell position Form factor of unpaired electrons in a given orbital (tabulated)

Atomic positions within the unit cell

Debye-waller factor (thermal motion of the ions)

o Spin-spin correlation function o spin components perp to Q (dipolar interaction) o w(k) for spin waves in close analogy with the case of phonons

Diffusion inélastique des neutrons Et excitations collectives

Phonons

Harmonic forces : Sum the forces Dynamics

Phonons

Monoatomic chain Fourier Transfrom Dispersion relation

Brillouin zone centers

Brillouin zone boundary

Phonons

Di-atomic chain Fourier transform: for each k, eigenvalue problem:

Acoustic mode In phase Vibrations

Optical mode Out of phase Vibrations

Phonons

General case Normal coordinates: Fourier transform: for each k, eigenvalue problem:

Dynamical matrix dimension

Longitudinal transverse

o Square of a frequency o associated « direction » (polarisation)

Phonons

o Phonons are bosons

o « detailed balance »

o Dynamical structure factor

o Selection rule

Bose-Einstein

Cross section

Creation Annihilation

CaF2

CaF2

CaF2

CaF2

CaF2

CaF2

Remember

is periodic. It suffices to study it in a single Brillouin zone

! These points are equivalent regarding the dispersion relation But have different neutron cross sections

Remember

Longitudinal phonon s :

t k

Q

Brillouin zone centers t = Brillouin zone node = Bragg peak position k = wavevector within the Brillouin zone

Remember

t

k

Q

Brillouin zone centers

Transverse phonon s :

Summary

The dispersion of phonons can be measured in (Q,w) space by means of inelastic neutron scattering With the help of a model, it becomes possible to measure physical parameteres as force constants

Spin excitation

Exchange : Heisenberg Hamiltonian

Spin excitation

Exchange : Heisenberg Hamiltonian Spins ordering below Néel temperature

Spin excitation

Exchange : Heisenberg Hamiltonian Spins ordering below Néel temperature Equation of motion

Cross section

Cross section

Creation Annihilation

o Spin waves are almost bosons

o « Detailed balance »

o Dynamical structure factor

Ferromagnet

Neutron cross section From quantization of the equation of motion

Ferromagnet

Static response (Bragg peak) With a different geometrical factor (best seen if Qz = 0, in contrats with the inelastic response)

Detailed balance

Dirac functions along the dispersion

Dynamical structure factor (Form factor, Debye, geometry)

Ferromagnet

Detailed balance

Dirac functions along the dispersion

Dynamical structure factor (Form factor, Debye, geometry)

Ferromagnet

Real space

Reciprocal space

t = Bragg peak position k = wavevector in the first Brillouin zone

Ferromagnet

!

Real space

Reciprocal space

These points are equivalent regarding the dispersion relation But have different neutron cross sections

Ferromagnet

Dispersion relation

Intensity superimposed on the dispersion relation, depends on the neutron cross section

Antiferromagnet

Dispersion relation

Intensity superimposed on the dispersion relation, depends on the neutron cross section

Summary

The dispersion of spin waves can be measured in (Q,w) space by means of inelastic neutron scattering With the help of a model, it becomes possible to measure physical parameteres as exchange constants

Example 1

Spin dynamics in LaSrMnO3

Example : manganites

Jc

Jab

Jc ~ 0.5 meV

Jab ~ -0.8 meV

Example : manganites

In the metallic state : spin wave in a metal ?

Example 2

Spin dynamics in triangular lattice

Example : triangular lattice

J

2

3

1

120° Néel order 3 spins per unit cell : 3 branches

1, 2 : correlations between in plane spin components 3 : correlations between out of plane spin components

Example : triangular lattice

2

3

1

Gap J

3 spins per unit cell : 3 branches

1, 2 : correlations between in plane spin components 3 : correlations between out of plane spin components

Example : triangular lattice

J

y

z

Q perpendicular to the hexagonal (yz) plane allows observing the branches corresponding to correlations between in plane spin components

Example : triangular lattice

J

z

y Q in the (yz) hexagonal case, restores intensity on the branch corresponding to correlations between in plane spin components

Example : triangular lattice

0

5

10

15

20

0 0.1 0.2 0.3 0.4 0.5

(q,0,0)

Deduce microscopic parameters from experiments J = 2.5 meV D = 0.5 meV

YMnO3

Example : triangular lattice

YMnO3 YbMnO3

Example : 1D magnets

Example 4

Spin dynamics in 1D systems

Example : 1D magnets

Dans certains systèmes, les fluctuations empêchent l’ordre magnétique de s’installer. Les excitations ne sont plus du tout des ondes de spin … Frustration, basse dimension, spin quantique =1/2, 1

Example 5

Crystal field

Anisotropy of magnetic moments

Example 6

Ondes de spin dans Un système géométriquement frustré Er2Ti2O7

Er2Ti2O7

XY spins & AF interactions (J > 0)

Extensive degeneracy

Archetype of magnetic « geometric » frustration :

the pyrochlore lattice (Re2Ti2O7)

« y3 »

Palmer Chalker

states

Er2Ti2O7

« y2 »

« y1 »

y2 is selected among degenerate XY states.

Why ? Is it « Order by disorder » ?

IN5 Xcrystal mode Full S(Q,w) in 2

days

« However »

One l One T One H

H= 2.5T (b) H=0 (a)

S(Q,w) calculated in RPA

References

[1] P.W. Anderson, Phys. Rev. 83, 1260 (1951)

[2] R. Kubo, Phys. Rev. 87, 568 (1952)

[3] T. Oguchi, Phys. Rev 117, 117 (1960)

[4] D.C. Mattis, Theory of Magnetism I, Springer Verlag, 1988

[5] R.M. White, Quantum Theory of Magnetism, Springer Verlag, 1987

[6] A. Auerbach, Interacting electrons and Quantum Magnetism, Springer

Verlag, 1994.