x-ray non-resonant and resonant magnetic...
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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
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European School on Magnetism
The Diamond synchrotron
3 GeV, 300 mA
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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.
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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.
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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.
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L. C. Chapon 6European School on Magnetism
Dipole radiation and forward emitting cone with relativistic e-
Poynting vector:
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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
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L. C. Chapon 8European School on Magnetism
Synchrotron brilliance versus transistor/inch2
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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
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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
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L. C. Chapon 11European School on Magnetism
Circular polarisation
k
k
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L. C. Chapon 12European School on Magnetism
F0
dW
ki
kf
Scattering X-ray
Transition rate, second order perturbation theory:
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L. C. Chapon 13European School on Magnetism
Maxwell's equations
In the absences of charges and currents (free space), the solutions are plane-waves:
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L. C. Chapon 14European School on Magnetism
Quantizing the free electromagnetic field
It is convenient to work in the Coulomb (radiation) Gauge, by setting:
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L. C. Chapon 15European School on Magnetism
Quantizing the free electromagnetic field
The Gauge condition implies: transverse waves with two orthogonal polarizationstates.
Electromagnetic energy:
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L. C. Chapon 16European School on Magnetism
Quantizing the free electromagnetic field
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L. C. Chapon 17European School on Magnetism
Electromagnetic field potential for a charge
Lorentz force is not conservative (depends on speed)
Using :
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L. C. Chapon 18European School on Magnetism
Hamiltonian of a charged particle in an EMF (no spin)
Generalized momentum:
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L. C. Chapon 19European School on Magnetism
Zeeman & Spin-orbit coupling terms
See Thomas factor for examplein Jackson “Classical Electromagnetism”
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L. C. Chapon 20European School on Magnetism
Full Hamiltonian charge in an EMF
Zeeman
SO coupling
Kinetic
Coulomb
EMF self-energy
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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
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L. C. Chapon 22European School on Magnetism
First and second order terms
Annihilatephoton first
Createphoton first
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L. C. Chapon 23European School on Magnetism
Non-resonant contribution of second-order terms
Using
Need to evaluate 4 commutators, with the help of :
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L. C. Chapon 24European School on Magnetism
Magnetic contribution
Here, lj and sj are given in units of
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L. C. Chapon 25European School on Magnetism
Separation of L and S
ss’pp’
2q
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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.
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European School on Magnetism R. Johnson , Phys. Rev. Lett. (2011)
Non-resonant magnetic scattering Cu3Nb2O8
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European School on Magnetism
P(c)
�1=( δ , δ ,0)
BiFeO3 : Non resonant micro-focused magnetic diffraction
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European School on Magnetism
50 mm resolution
BiFeO3 : Non resonant micro-focused magnetic diffraction
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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
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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
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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
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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
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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.
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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
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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
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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.
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L. C. Chapon 38European School on Magnetism
Note on link between cross section and absorption spectroscopy
Optical theorem
XMCD
XMLD
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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)
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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.
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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