X-ray non-resonant and resonant magnetic...

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

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Circular polarisation

k

k

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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:

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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 :

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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

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First and second order terms

Annihilatephoton first

Createphoton first

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Non-resonant contribution of second-order terms

Using

Need to evaluate 4 commutators, with the help of :

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Magnetic contribution

Here, lj and sj are given in units of

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Separation of L and S

ss’pp’

2q

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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'

qq

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

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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