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L. C. Chapon 1European School on Magnetism
X-ray non-resonant and resonant magnetic scattering Laurent C. Chapon, Diamond Light Source
European School on Magnetism
The Diamond synchrotron
3 GeV, 300 mA
L. C. Chapon 3European School on Magnetism
Lienard-Wiechert potentials
r : position of the observer
rq : position of the charge
The retarded time tr:
n.b: Use S.I units throughout.
L. C. Chapon 4European School on Magnetism
Lienard-Wiechert potentials
By changing variable: and considering the Jacobian, one finds:
with
The difficulty is in evaluating the vector fields at time t. This involves derivatives with a lot of chain rules.
L. C. Chapon 5European School on Magnetism
Fields
The far-field part of the electric field is:
One can also prove that:
These expressions are evaluated at the retarted time tr.
L. C. Chapon 6European School on Magnetism
Dipole radiation and forward emitting cone with relativistic e-
Poynting vector:
L. C. Chapon 7European School on Magnetism
Brilliance and polarisation
BendingMagnet
InsertionDevice
Very small angular opening of the forward emittance cone (fraction of a mrad)
Highly polarised radiationin the synchrotron orbit plane
L. C. Chapon 8European School on Magnetism
Synchrotron brilliance versus transistor/inch2
L. C. Chapon 9European School on Magnetism
Conventions : Vertical geometry
s
s’pp’
kk’
2q
Q s and p refer to linear polarisation of light perpendicular and parallel to the scattering plane
L. C. Chapon 10European School on Magnetism
Conventions: Horizontal geometry
ss’ p
p’ kk’2q Q
s and p refer to linear polarisation of light perpendicular and parallel to the scattering plane
L. C. Chapon 11European School on Magnetism
Circular polarisation
k
k
L. C. Chapon 12European School on Magnetism
F0
dW
ki
kf
Scattering X-ray
Transition rate, second order perturbation theory:
L. C. Chapon 13European School on Magnetism
Maxwell's equations
In the absences of charges and currents (free space), the solutions are plane-waves:
L. C. Chapon 14European School on Magnetism
Quantizing the free electromagnetic field
It is convenient to work in the Coulomb (radiation) Gauge, by setting:
L. C. Chapon 15European School on Magnetism
Quantizing the free electromagnetic field
The Gauge condition implies: transverse waves with two orthogonal polarizationstates.
Electromagnetic energy:
L. C. Chapon 16European School on Magnetism
Quantizing the free electromagnetic field
L. C. Chapon 17European School on Magnetism
Electromagnetic field potential for a charge
Lorentz force is not conservative (depends on speed)
Using :
L. C. Chapon 18European School on Magnetism
Hamiltonian of a charged particle in an EMF (no spin)
Generalized momentum:
L. C. Chapon 19European School on Magnetism
Zeeman & Spin-orbit coupling terms
See Thomas factor for examplein Jackson “Classical Electromagnetism”
L. C. Chapon 20European School on Magnetism
Full Hamiltonian charge in an EMF
Zeeman
SO coupling
Kinetic
Coulomb
EMF self-energy
L. C. Chapon 21European School on Magnetism
Perturbing Hamiltonian
Refers to Blume, 1985
● Terms square in A contribute to the first order scattering term ● Terms linear in A contribute to the second order scattering term
L. C. Chapon 22European School on Magnetism
First and second order terms
Annihilatephoton first
Createphoton first
L. C. Chapon 23European School on Magnetism
Non-resonant contribution of second-order terms
Using
Need to evaluate 4 commutators, with the help of :
L. C. Chapon 24European School on Magnetism
Magnetic contribution
Here, lj and sj are given in units of
L. C. Chapon 25European School on Magnetism
Separation of L and S
ss’pp’
2q
L. C. Chapon 26European School on Magnetism
Summary non-resonant magnetic X-ray scattering
● Rest mass of the electron =511 keV● At 1 keV, scattering cross section 3.8 10-6 smaller than Thomson scattering. ● Only a few unpaired electrons contribute to the magnetic scattering vs. all electrons
in the Thomson scattering. ● However, the flux available largely compensate for
the weak scattering cross section.
European School on Magnetism R. Johnson , Phys. Rev. Lett. (2011)
Non-resonant magnetic scattering Cu3Nb2O8
European School on Magnetism
P(c)
�1=( δ , δ ,0)
BiFeO3 : Non resonant micro-focused magnetic diffraction
European School on Magnetism
50 mm resolution
BiFeO3 : Non resonant micro-focused magnetic diffraction
European School on Magnetism
ki k
f
Q hs
p
s'
p'
L R
Experiment off resonance, 5.8 KeV
R. D. Johnson,et al., Phys. Rev. Lett. 110, 217206 (2013)
BiFeO3 : Non resonant micro-focused magnetic diffraction
European School on Magnetism
k1=(d,d,0)
c
k2=(d,-2d,0) k
3=(-2d,d,0)
k1=(d,d,0) k
2=(d,-2d,0) k
3=(-2d,d,0)
c c
P P P
PP P
BiFeO3 : “Homochiral” domains
European School on Magnetism
● Tune the energy of the X-ray beam to an absorption edge
● Photon is absorbed and the systemre-emit a photon with the same energy
● Element specific information● Combine the element specific
information from spectroscopy techniques with Bragg scattering
● Polarization dependence effects● Possibility to probe magnetic order
but also other E/M-multipoles
Resonant X-ray magnetic scattering
L. C. Chapon 33European School on Magnetism
Resonant X-ray magnetic scattering
We need to evaluate the matrix elements:
The power expansion is justified by the fact that the spatial extend of the core electron is relatively small (~0.1 A)
We also use the following commutator, to switch to position representation
E1 E2E3
M1 M2M3
L. C. Chapon 34European School on Magnetism
Resonant X-ray magnetic scattering : E1-E1
[1] J. P. Hill and D. F. McMorrow, “X-ray resonant exchange scattering: polarization dependence and correlation functions,” Acta Crystallogr., vol. A52, pp. 236–244, 1996.
European School on Magnetism
Incommensurateq=(1/2+d,0,0.2)q=(1/2,0,0)
N. Lee, Phys. Rev. Lett. 110, 137203 (2013)
Resonant X-ray magnetic scattering GdMn2O5
European School on Magnetism
LIII Gd
Off-resonance
N. Lee, Phys. Rev. Lett. 110, 137203 (2013)
a
b
Pb
Pb
Resonant X-ray magnetic scattering GdMn2O5
E1-E1
L. C. Chapon 37European School on Magnetism
Probing different multipolar orders
[1] S. Di Matteo, “Resonant x-ray diffraction: multipole interpretation,” J. Phys. D. Appl. Phys., vol. 45, no. 16, p. 163001, 2012.
L. C. Chapon 38European School on Magnetism
Note on link between cross section and absorption spectroscopy
Optical theorem
XMCD
XMLD
L. C. Chapon 39European School on Magnetism
Magnetic X-ray holography
Thomas A. Duckworth, Feodor Ogrin, Sarnjeet S. Dhesi, Sean Langridge, Amy Whiteside, Thomas Moore, Guillaume Beutier, and Gerrit van der Laan, "Magnetic imaging by x-ray holography using extended references," Opt. Express 19, 16223-16228 (2011)
European School on Magnetism
Ptychography
[1] C. Donnelly et al., “Three-dimensional magnetization structures revealed with X-ray vector nanotomography,” Nature, vol. 547, no. 7663, pp. 328–331, 2017.
L. C. Chapon 41European School on Magnetism
● Born approximation valid● Magnetic ~ nuclear scattering amplitude ● Very little beam heating low T→
● Large penetration depth, bulky sample environments (magnets, dilution….)
● Manipulate polarization and analysis but costly (flux)
● Large divergence, relatively poor Q-resolution
● Lack of spatial resolution
● Flux typically up to 1010 n.cm-2.s-1
(scattering volume)
● No direct L/S separation (only by fitting form factor)
Neutrons and X-ray for magnetic scattering
Neutron X-ray
● Off resonance can get quantitative M but scaling to charge scattering not always easy(use of attenuators for charge scattering...)
● Magnetic Xs much smaller but compensated by flux.
● Beam heating can be a problemnot straightforward to go to dilution T
● Not easy to do k=0 work
● Manipulate polarization and analysis
● Highly collimated, excellent Q-resolution
● Spatial resolution down to 20nm
● High brilliance and flux● Direct L/S separation● Resonant element specific →● Resonant probe tensor beyond →
magnetic dipole