Post on 19-Oct-2020
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.